Use a computer regression package, to work these two computer exercises.
2. Ozark Bottled Water Products, Inc. hired a marketing consulting firm to perform a
test marketing of its new brand of spring water called Liquid Ozarka. The
marketing experts selected 15 small and medium-sized towns in Arkansas and
Missouri for a one-month-long sales test. For one month, Liquid Ozarka was sold
at a variety of prices ranging from $3 per gallon to $4 per gallon. Specifically, in
three of the markets, price was set by the marketing experts at $3 per gallon. In
three more markets, price was set at $3.25 per gallon, and so on. The prices
charged in each market (P) are shown in the table below. For each of the 15 market
areas, the marketing consultants collected data on average household income (M),
the population of the marketing area (N), and the price of a rival brand of bottled
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water (PR). At the end of the month, total sales of Liquid Ozarka (Q) were
tabulated to provide the following data from which the consultants estimated an
empirical demand function for the new product.
Market P M PR N Q
1 $3.00 $45,586 $2.75 274,000 7,952
2 3.00 37,521 3.50 13,450 8,222
3 3.00 41,333 2.64 54,150 7,166
4 3.25 47,352 2.35 6,800 6,686
5 3.25 51,450 2.75 11,245 7,715
6 3.25 27,655 3.15 54,500 6,643
7 3.50 30,265 2.55 26,600 5,155
8 3.50 39,542 3.00 158,000 7,127
9 3.50 41,596 2.75 22,500 5,834
10 3.75 42,657 2.45 46,150 5,093
11 3.75 36,421 2.89 8,200 5,828
12 3.75 47,624 2.49 38,500 6,590
13 4.00 50,110 3.15 105,000 6,228
14 4.00 57,421 2.80 92,000 7,218
15 4.00 38,450 2.90 38,720 5,846
Using the marketing data from the 15 test markets shown above, estimate the
parameters of the linear empirical demand function:
Q = a + bP + cM + dPR + eN
If any of the parameter estimates are not significant at the 2 percent level of
significance, drop the associated explanatory variable from the model and estimate
the demand function again.
a. Your estimated linear demand function for Liquid Ozarka is
ˆQ
= _______________________________________.
b. What percentage of the variation in sales of Liquid Ozarka is explained by
your estimated demand function?
The marketing consultants describe a “typical” market as one in which the price of
Liquid Ozarka is $3.50 per gallon, average household income is $45,000, the price
of rival bottled water is $3 per gallon, and the population is 75,000. Answer the
following questions for this “typical” market scenario.
c. What is the estimated elasticity of demand for Liquid Ozarka? Is demand
elastic or inelastic? What would be the percentage change in price required
to increase sales of Liquid Ozarka by 10 percent?
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d. What is the estimated income elasticity of demand? Is Liquid Ozarka a
normal or inferior good? A 6 percent increase in average household income
would be predicted to cause what percentage change in sales of Liquid
Ozarka?
e. What is the estimated cross-price elasticity of demand for Liquid Ozarka
with respect to changes in price of its rival brand of bottled water? Does the
estimated cross-price elasticity have the expected algebraic sign? Why or
why not? If the price of the rival brand of water rises by 8 percent, what is
the estimated percentage change in sales of Liquid Ozarka?
Using the marketing data from the preceding 15 test markets, estimate the
parameters for the log-linear empirical demand function:
Q = aPbMcPR
dNe
If any of the parameter estimates are not significant at the 2 percent level of
significance, drop the associated explanatory variable from the model and estimate
the demand function again.
f. Your estimated log-linear demand function for Liquid Ozarka is
ˆQ
= _______________________________________.
g. Does a log-linear specification work better than a linear specification of
demand for Liquid Ozarka? Explain by comparing F-ratios, R2s, and t-ratios
(or p-values).
h. Using the estimated log-linear demand function, compute the price, income,
and cross-price elasticities of demand. How do they compare to the estimated
elasticities for the linear demand specification?
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161
3. For 2007–2009, Gallaway, Inc. has collected the following data on monthly sales
of its Titan II driving club, where Q = the number of units sold per month.
Year Month Q Year Month Q Year Month Q
2007 January 6,942 2008 January 8,007 2009 January 7,925
February 7,348 February 7,698 February 7,326
March 7,328 March 7,417 March 8,037
April 8,350 April 8,897 April 9,087
May 8,619 May 8,607 May 9,303
June 9,282 June 9,314 June 9,139
July 8,183 July 8,686 July 8,105
August 8,317 August 8,539 August 8,321
September 8,552 September 8,967 September 8,960
October 7,993 October 8,507 October 7,580
November 8,198 November 8,359 November 8,562
December 8,082 December 8,157 December 8,072
a. Management at Gallaway is concerned about sales. They would like to know
if there is an upward trend is sales of the Titan II. Use the data above to
estimate the monthly trend in sales using a linear trend model of the
form: Qt = a + bt . Does your statistical analysis indicate a trend? If so, is it
an upward or downward trend and how great is it? Is it a statistically
significant trend (use the 5 percent level of significance)?
b. Now adjust your statistical model to account for seasonal variation in club
sales. Estimate the following model of sales:
Qt = a + bt + c1D1t + c2D2t + c3D3t
where D1t = 1 for the months of January–March or 0 otherwise, D2t = 1 for
the months of April–June or 0 otherwise, and D3t = 1 for the months of
July–September or 0 otherwise. Do the data indicate a statistically significant
seasonal pattern (use the 5 percent level of significance)? If so, what is the
seasonal pattern of sales of Titan II clubs?