Estimation of Profit-Maximizing Price for One Round of Golf Econ 382 Ellis, GregAlexander F. Woodhouse, Shiqiu Wu, Qiong Hu

Executive Summary:

In this project, we are given data from 22 golf courses in the greater Seattle area. We

want to find the demand function for golf courses in terms of the price, given all of the

other explanatory variables. Our goal is to find the best estimation of the demand function

for the MGS courses given the sample provided. In addition to giving advice to MGS as to

whether or not they should implement the seasonal green fee, we also want to find out if

they should fund the paving of cart paths or the building of driving ranges first, with the

interest of increasing their revenue.

First, we make our educated guess for the relationship of each variable on ROUNDS.

Then we analyze the unrestricted model by looking at p-values. Since WINTER*FEE, YARD

and RANGE have insignificant p-values, we use Wald Test to confirm that they are

irrelevant variables and remove them. When we get the improved model, we use White

Test to detect the Heteroskedasticity and adjust our current model for it. By now, each

variable has a small p-value and the adjusted R square increases. Then we analyzed the

sign and magnitude of each variable and found they are all reasonable and logical. Finally,

we conclude this model is best linear unbiased estimates. To give our formal analysis to the

MGS courses, we computed and compared price elasticity of demand in winter time and in

non-winter time, and concluded that MGS should use seasonal green fees since P.E.D. in

winter is more elastic than in non-winter time. Also, we suggest that MGS should build cart

paths through all three of their courses before building driving ranges.

Formal Analysis:

The nonprofit organization Municipal Golf of Seattle (MGS) currently manages three

golf courses owned by the city of Seattle: Jackson, Jefferson, and West Seattle. As their

economic advisers, we have been hired to advise them if, as well as how much, they should

reduce their greens fees during the winter months, as well as helping them decide where to

allocate additional funds for capital improvements. Because MGS works with the city of

Seattle and all costs are handled by the state, our entire analysis will be under the

assumption that total costs are zero, and therefore total profit is equal to total revenue. It

follows that in determining how we will advise MGS, our goal will be to maximize the total

revenue, subject to a number of constraints.

To help us answer these two questions, we have been given access to data from 22

golf courses in the greater Seattle area, measured once a month over a one-year span. For

each of these 268 data points, the total number of times the course was played during the

given month at the given course (ROUNDS) is given, along with a number of possible

explanatory variables corresponding to the data point in question. To investigate the roles

of these possible explanatory variables on the dependent variable ROUNDS, we first made

an educated guess about their relationship with ROUNDS as follows:

ROUNDS=f (FEE ,FEESUB ,RAIN ,TEMP ,RATING ,SLOPE ,YARD , DIS ,CART ,RANGE ,WINTER ,MGS )(− ,+ ,−,+ ,+ ,− ,? ,−,+ ,? ,−, ? )

In our attempt to maximize the total revenue, it will be represented by the product:

ROUNDS∗FEE, subject to the constraints of the values of the other relevant explanatory

variables present. For the first question, we will examine the elasticity of demand at the

MGS courses in the winter and non-winter, and if applicable, find the value of FEE that

maximizes total revenue during winter. But before we can determine either of these things,

we need to use the data to estimate the best possible model for the demand of ROUNDS.

Among the possible explanatory variables listed above is FEE, which is the average

price charged by the given course for each round of golf played in a given month. We expect

that FEE would have a negative relationship with ROUNDS, which corresponds to the law of

demand in economic theory. To determine the degree and magnitude of this relationship,

we need to develop a model that includes the effects of the other relevant variables as well.

If we can verify that certain assumptions have been met, then we can use the ordinary least

squares (OLS) method and know that it will provide the best linear unbiased estimates of

this relationship.

An unrestricted model (Figure 1) is available as a potential estimation model,

however, it is doubtful that it is the best fit. Specifically, we suspect that there could be at

least one irrelevant variable in the model. Running t-Tests with the null hypothesis

H 0 : βk=0 against the alternative hypothesis H a : βk≠0 for each of the independent variables

(k=1 ,2 ,…,17 ), we see that there is very high probability that the null hypothesis cannot be

rejected for the coefficients of YARD, RANGE, and WINTER*FEE. To test if these three

variables can be removed from the model, we run a Wald (F) Test with the null hypothesis

H 0 : βYARD=βRANGE=βWINTER∗FEE=0 against the alternative hypothesis

H A :at least oneof the βs isnot equal ¿zero. We calculated that the observed

F stat=0.102417<F3 ,246¿ (0.05 )=2.60, and so we fail to reject the null hypothesis (Figure 2).

This means that we can improve the fit of our model by eliminating these irrelevant

variables, at the 5% level of significance.

After we eliminated those irrelevant variables, we get a restricted model without

irrelevant variables (WINTER*FEE, YARD, and RANGE) (Figure 3). Now we want to

determine whether or not there is heteroskedasticity in our model. If there is, then we

must eliminate it if we want to be certain that we are attaining the best linear unbiased

estimates. To test for heteroskedasticity in our model, we used the White test, primarily

because it has the ability to test for a wider range of forms of heteroskedasticity, making it

more comprehensive and precise than the Park test. We performed the White test with the

null hypothesisH 0 :α 1=α 2=…=α 90=0 against the alternative hypothesis

H a :at least one of the α values is not equal ¿ zero, and we obtain a statistic equal to

obs∗R2=268∗R2=131.2916 ,which is larger than the critical value which is

χ902 (0.05 )=113.145. This result means that we must reject the White Test at a significance

level of 5%, which leads us to conclude that there is heteroskedasticity in our current

model. To fix the heteroskedasticity, we must adjust the Standard Errors of each error term

to account for it. It shows the Standard Errors for coefficients decreased after our

adjustment (Figure 5).

Now that we have adjusted our model for heteroskedasticity, we can be confident

that we have found a model for ROUNDS that has homoscedasticity. To be sure that this

model is realistic and logical, we must verify that the respective signs and magnitudes of

the coefficients for each characteristic match our initial intuition. The first thing we notice

is that the coefficient for FEE is positive, which contradicts both our intuition and the law of

demand. For the purposes of our analysis, we are only concerned with the demand model

pertaining to the MGS courses. In terms of our analysis, this means that we only care about

the condition of the model in which mgs=1. Under this condition, the total effect of FEE on

ROUNDS is always negative because the coefficient corresponding to MGS*FEE (which is

calculated to be -337.5988) offsets the positive effect of FEE on ROUNDS (βFEE+β FEE∗MGS<0 )

. The coefficient of FEESUB is consistent with our intuition, and shows a positive

relationship between the fee charged by other courses and the demand for a specific

course. This is also in correspondence the effect of the price of a substitute on demand in

economic theory. The coefficient of RATING is also positive, which matches our intuition,

which was that the courses with higher ratings in terms of difficulty would attract better

golfers, but not necessarily detract golfers with less skill. On the other hand, the negative

coefficient in front of SLOPE matches our intuition because less skilled golfers would prefer

a course with less hills, but remain ambivalent about the rating. WINTER has a negative

relationship with ROUNDS, and the coefficient in front of WINTER has a very large

magnitude (βWINTER=¿-1586.647), which matches our intuition as well because days are

shorter during the winter, and so there are less available tee times, in addition, the weather

conditions are not as favorable. We estimated that RAIN is a more important factor than

TEMP. RAIN has a bigger absolute coefficient value than TEMP has, which makes sense

because even the temperature was comfortable for golfing, people still would be less likely

to go golfing if it was rainy. With regards to the effects of TEMP, we would expect positive

correlation with ROUNDS because golf is generally more popular in areas with warmer

climates. Finally, the coefficients of CART*WINTER and DISTANCE interpret that people

would go to the golf course with cart path more, and have less preferences on the courses

which are further from Seattle. Besides, each variable has a small (significant) p value, and

the sign of each coefficient is reasonable and logical. Also, the adjusted R squared value

increased to R2=¿0.865925 which is higher than 0.864380, the previous value it had in the

unrestricted model. Now that we have verified that the implications of our model hold true

in terms of logic and intuition, we can conclude that this model (figure 5) provides the best

linear unbiased estimates.

Based on this restricted model, we can provide some consulting advice to MGS

regarding the structure of their greens fees, specifically in the winter months. We

calculated the price elasticity of demand for MGS golf courses during winter and non-

winter time to determine if MGS should implement the seasonal rate. We calculated the

price elasticity of demand in wintertime (WINTER=1) to be η=−2.220695, and

η=−0.546797 as the price elasticity of demand during non-winter time (WINTER=0). Since

(-2.220695)<-1, it tells us that it is elastic during winter, which means a small change in

price causes a larger change in quantity. Also, the price elasticity of demand in winter is

more elastic than the one in non-winter time because |-2.220695|>|-0.546797|. According

to this result, we recommend MGS to use seasonal rates, specifically by lowering their

greens fees in the winter. SinceTotal Revenue=TR=P∗Q=FEE∗ROUNDS, if the price

elasticity of demand is elastic, we can increase the total revue by decreasing the value of

FEE. Specifically, MGS will maximize their revenue, and therefore their profits, by lowering

their winter-time greens fees to $12.20 (complete derivation in appendix B).

To consider whether to build the cart path or the driving range first, we go back to

look at the unrestricted model since it includes both CART*WINTER and RANGE. We found

CART*WINTER has much larger coefficient value than RANGE, so this means cart has larger

impact on increasing ROUNDS than driving range. Therefore, we suggest MGS to fund the

paving of cart paths first.

Appendix A

Figure 1 Unrestricted Model

Figure 2 Wald Test (F-test)

Figure 3 Restricted Model

Figure 4 White Test (Heteroskedasticity)

... (omission of extraneous data)…

Figure 5 Best Fit Model

Appendix B

Derivation of the Profit-Maximizing Fee during the Winter:

NOTE : throughout thisderivation ,assume FEE=p ,∧ROUNDS (FEE )=q ( p )

Profit=π (p )=TR (p )−TC=TR ( p ) , sinceTC=0

Total Revenue=TR ( p )=[q (p ) ] ∙ p

q ( p )=β0−β1 p−β2 p2

π ( p )=TR ( p )=[β0−β1 p−β2 p2 ] ∙ p=β0 p−β1 p

2−β2 p3

max{p }

[ π (p ) ] : ∂∂ p

[ π ( p¿) ]=0 , ∂2

∂ p2[π ( p¿ ) ]<0

∂∂ p [π ( p ) ]=β0−2β1 p−3 β2 p

2=0

p=−2 β1±√(2β1 )2−4 (3 β2 ) (−β0 )

2 (3 β2 )=

−β1(3 β2 )

± √(2 β1 )2−4 (3 β2) (− β0 )2 (3β2 )

∂2

∂ p2[π ( p ) ]=−2β1−6 β2 p<0

p¿>−β1(3 β2 )

so:

p¿=−β1(3β2 )

+ √(2 β1 )2−4 (3 β2 ) (−β0 )2 (3β2 )

by estimating β0 , β1 ,∧β2 with our regression coefficients (and average values of input

variables held constant) we find that:

p¿=$ 12.20