1. Estimation of Profit-Maximizing Price for One Round of Golf
Econ 382
Ellis, Greg
Alexander 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.
2. 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
3. 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
𝐻0: 𝛽 𝑘 = 0 against the alternative hypothesis 𝐻 𝑎: 𝛽 𝑘 ≠ 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 𝐻0: 𝛽 𝑌𝐴𝑅𝐷 = 𝛽 𝑅𝐴𝑁𝐺𝐸 = 𝛽 𝑊𝐼𝑁𝑇𝐸𝑅 ∗𝐹𝐸𝐸 = 0 against the alternative
hypothesis 𝐻𝐴: 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝛽𝑠 𝑖𝑠 𝑛𝑜𝑡 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑧𝑒𝑟𝑜. We calculated that the
observed 𝐹𝑠𝑡𝑎𝑡 = 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.
4. 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 hypothesis 𝐻0: 𝛼1 = 𝛼2 = ⋯ = 𝛼90 = 0 against the alternative hypothesis
𝐻 𝑎: 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝛼 𝑣𝑎𝑙𝑢𝑒𝑠 𝑖𝑠 𝑛𝑜𝑡 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑧𝑒𝑟𝑜, and we obtain a statistic equal to
𝑜𝑏𝑠 ∗ 𝑅2
= 268 ∗ 𝑅2
= 131.2916, which is larger than the critical value which is
𝜒90
2 (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 errorterm
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
5. 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 ( 𝛽 𝐹𝐸𝐸 +
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 (𝛽 𝑊𝐼𝑁𝑇𝐸𝑅 =-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 𝑅̅2
=0.865925 which is higher than 0.864380, the previous value it had
6. 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. Since 𝑇𝑜𝑡𝑎𝑙 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.
