Question

Below is a case on estimation and analysis of demand for Bottled Water.

Read the case carefully and use the appropriate techniques given in the text book on demand estimation and analysis and make your decisions, judgments and evaluation based on the results. For solving any part of the case you have to give your explanations write the proper formula, and show the procedure of reaching to your answers. All your work should be typed and the data, printout of all regressions output should be annexed for reference.

Estimation and Analysis of Demand for Ozarka Bottled Water Products

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 water (P_R). 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

1. Using the marketing data from the 15 test markets shown above, estimate the parameters of the linear empirical demand function:  ?=?+??+??+???+??

1. Using the appropriate tests, Evaluate the significance of the regression as a whole and the significance of the individual parameters at 5% level of significance. (show all your steps and procedure).

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.

1. Your final estimated linear demand function for Liquid Ozarka is

Q=_______________________________________.

2. What percentage of the variation in sales of Liquid Ozarka is explained by your estimated demand function?

2. 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.

1. 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?

2. 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?

3. 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?

2. Using the marketing data from the preceding 15 test markets, estimate the parameters for the log-linear empirical demand function:

?=??b?c??d?e

1. Using the P-Values, 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.

3. Your final estimated log-linear demand function for Liquid Ozarka is

Q=_______________________________________.

2. 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).

3. Using the estimated log-linear demand function, compute the price, income, and cross-price elasticity of demand. How do they compare to the estimated elasticity for the linear demand specification?
4. What is the estimated quantity demand based on the values of explanatory variables given above?

Answer 1

Answer;

Let P be the price in market

M be average household income

N be population of the marketing area

PR be the price of a rival brand of bottled water

Q be total sales of Liquid Ozarka

Here Q is dependent variable and remaining are independent variables.

There are 15 observations each.

Here the problem is of multiple regression.

We can do multiple regression in minitab.

steps :

ENTER data into MINITAB sheet --> STAT --> Regression --> Regression --> Response : Q --> Predictors : Selecy remaining variables --> Results : select second option --> ok --> ok

Welcome to Minitab, press F1 for help.

Regression Analysis: Q versus P, M, PR, N

The regression equation is
Q = 5387 - 1853 P + 0.0767 M + 1562 PR + 0.00140 N

Predictor Coef SE Coef T P
Constant 5387 1644 3.28 0.008
P -1853.3 318.7 -5.81 0.000
M 0.07665 0.01549 4.95 0.001
PR 1562.0 378.2 4.13 0.002
N 0.001404 0.001620 0.87 0.406

S = 412.631 R-Sq = 86.8% R-Sq(adj) = 81.6%

Analysis of Variance

Source DF SS MS F P
Regression 4 11231314 2807829 16.49 0.000
Residual Error 10 1702646 170265
Total 14 12933960

The estimated linear demand function for Liquid Ozarka is,

Q = 5387 - 1853 P + 0.0767 M + 1562 PR + 0.00140 N

The percentage of the variation in sales of Liquid Ozarka is explained by your estimated demand function is 86.8%.

Assume alpha = significance level = 2% = 0.02

Here we have to test the hypothesis that,

H0 : Bj = 0 Vs H1 : Bj not= 0

where Bj is population slope for jth independent variable.

Here test statistic follows F-distribution.

This test is also known as overall significance test.

Here test statistic is 16.49

ANd P-value is 0.000

P-value < alpha

Reject H0 at 2% significance level.

Atleast one of the slope is differ than 0.

By using t-test N is insignificant variable since p-value > alpha.

Therefore N is deleted from the model and remaining variables are included into the model.

When we deleted N from the model then the model is,

The regression equation is
Q = 5357 - 1907 P + 0.0803 M + 1616 PR

Predictor Coef SE Coef T P
Constant 5357 1625 3.30 0.007
P -1906.9 309.2 -6.17 0.000
M 0.08033 0.01473 5.45 0.000
PR 1615.7 368.9 4.38 0.001

S = 407.955 R-Sq = 85.8% R-Sq(adj) = 82.0%

Analysis of Variance

Source DF SS MS F P
Regression 3 11103262 3701087 22.24 0.000
Residual Error 11 1830699 166427
Total 14 12933960

Now the model is significant.

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.

Here we have to predict Q when P = $3.50, M = $45,000, PR = $3, and N = 75,000

The regression equation is,

Q = 5387 - 1853 P + 0.0767 M + 1562 PR + 0.00140 N

Q = 5387 - 1853*3.50 + 0.0767*45000 + 1562*3 + 0.00140*75000 = 7144

Here Coefficient of price = -1853

Interpretation : When we fixed M, PR and N then one unit change in P will be 1853 unit decrease in Q.

Using the marketing data from the preceding 15 test markets, estimate the parameters for the log-linear empirical demand function:

For this we have to take natural logarithm of each of the variable and estimate the regression equation.

Regression Analysis: ln(Q) versus ln(P), ln(M), ln(PR), ln(N)

The regression equation is
ln(Q) = 4.17 - 0.984 ln(P) + 0.474 ln(M) + 0.679 ln(PR) + 0.0106 ln(N)

Predictor Coef SE Coef T P
Constant 4.169 1.143 3.65 0.004
ln(P) -0.9844 0.1867 -5.27 0.000
ln(M) 0.4737 0.1052 4.50 0.001
ln(PR) 0.6792 0.1889 3.60 0.005
ln(N) 0.01061 0.01824 0.58 0.574

S = 0.0712864 R-Sq = 83.3% R-Sq(adj) = 76.6%

Analysis of Variance

Source DF SS MS F P
Regression 4 0.253750 0.063437 12.48 0.001
Residual Error 10 0.050817 0.005082
Total 14 0.304567

Here also we get same result about overall significance.

ANd using individual significance we exclude ln(N) from the model because P-value(0.574) > alpha (0.02).

R-sq = 83.3%

The estimated log-linear demand function for Liquid Ozarka is ln(Q) = 4.17 - 0.984 ln(P) + 0.474 ln(M) + 0.679 ln(PR) + 0.0106 ln(N)