Managerial Economics Study Set 8

Business

Quiz 4 :

Estimating Demand

Quiz 4 :

Estimating Demand

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The county assessor (see Exercise) feels that the use of more independent variables in the regression equation might improve the overall explanatory power of the model. In addition to size, the assessor feels that the total number of rooms, age, and whether or not the house has an attached garage might be important variables affecting selling price. The data for the 15 randomly selected dwellings are shown in the following table. a. Using a computer regression program, determine the estimated regression equation with the four explanatory variables shown in the following table. b. Give an economic interpretation of each of the estimated regression coefficients. c. Which of the independent variables (if any) is statistically significant (at the.05 level) in explaining selling price d. What proportion of the total variation in selling price is explained by the regression model e. Perform an F-test (at the 5 percent significance level) of the overall explanatory power of the model. f. Construct an approximate 95 percent prediction interval for the selling price of a 15-year-old house having 1,800 square feet, 7 rooms, and an attached garage. img carefully examined. Demand forecasts usually rely on time-series data. In contrast, cross-section data appear in Table. Soft drink consumption in cans per capita per year is related to six-pack price, income per capita, and mean temperature across the 48 contiguous states in the United States. Exercise Cascade Pharmaceuticals Company developed the following regression model, using time-series data from the past 33 quarters, for one of its non-prescription cold remedies: Y = 1.04 + 0.24X1 0.27X2 whereY = quarterly sales ðin thousands of casesÞ of the cold remedy X1 = Cascade's quarterly advertising ð× $1,000Þ for the cold remedy X2 = competitors' advertising for similar products ð× $10,000) Here is additional information concerning the regression model: sb1 = 0:032 sb2 = 0:070 R 2 = 0:64 se = 1:63 F-statistic = 31:402 Durbin-Watson (d) statistic = 0.4995 a. Which of the independent variables (if any) appears to be statistically significant (at the 0.05 level) in explaining sales of the cold remedy b. What proportion of the total variation in sales is explained by the regression equation c. Perform an F-test (at the 0.05 level) of the overall explanatory power of the model. d. What additional statistical information (if any) would you find useful in the evaluation of this model TABLE Soft Drink Demand Data (available as an Excel file on this book's Web site) img
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a) The data below shows the selling price related to size, number of rooms, age, and provision of garage across 15 randomly selected dwellings.
img To run regression, select the "Data" tab and click "Data Analysis" in the "Analysis" grouping.
• Input the data of dependent variable in the "Input Y-Range" field, and then enter data of independent variables in the "Input X-Range" field for multiple columns.
• Click "OK" to create the analysis as below.
img The intercept is calculated as 185.265. The coefficients of size, number of rooms, age, and provision of garage are 3.92, 3.58,
img , and
img , respectively.
Thus, the regression equation estimated using OLS is
img .
b) According to the estimated regression equation:
• If the size of the house increases by 100 sq. ft., the price of housing increases by
img (keeping number of rooms, age and provision of garage unchanged).
• If the number of rooms increases by 1, the price of housing increases by
img (keeping size, age and provision of garage unchanged).
• If the age of a house increases by 1, the price of housing declines by
img (keeping size, number of rooms and provision of garage unchanged).
• If a house having an attached garage is estimated to be sold at
img less than a house without an attached garage (keeping size, number of rooms and age of the house unchanged).
c) The calculated
img -stat for size of the house (5.19) is quite high as compared to the table value
img . This implies that the coefficient of size of house significantly differ from zero at 0.05 level of significance.
However, the
img -stat for number of rooms, age and provision of garage are statistically insignificant in explaining the variations in selling price, at the 0.05 level of significance.
d) The value of adjusted
img square is
img which implies that the regression model explains around 85 percent of the total variation in the selling price.
e) Perform the following
img test to determine the statistical significance of the overall explanatory power of the regression model.
The null hypothesis is
img (no relationship between price and independent variables)
And, the alternative hypothesis is
img (relationship between price and at least one independent variable)
With 15 observations and 4 independent variables, a
img -distribution for each level of statistical significance has
img degrees of freedom for numerator and
img degrees of freedom for denominator.
(
img is the total number of observations and
img is the number of estimated regression parameters, including the constant term)
From Table 3 of Appendix B, the
img -ratio for the above mentioned degrees of freedom at the 5 percent level of significance is 3.48.
The
img statistic, however, in the regression model is calculated as 20.848. This value is quite greater than the critical value.
Thus, we reject the null hypothesis of no relationship between the selling price and independent variables. That is, the four explanatory variables explain a significant proportion of the variations in selling price, at the 5 percent level of significance.
f) Substitute
img ,
img ,
img , and
img in the estimated regression equation to obtain the price
img forecast.
img The standard error of the regression
img as calculated in part (a) is 11.135.
Thus, an approximate 95 percent forecast interval of
img is
img Thus, we are 95 percent confident that the true value of a 15-year-old house having 1,800 square feet, 7 rooms, and an attached garage will be
img .

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General Cereals is using a regression model to estimate the demand for Tweetie Sweeties, a whistle-shaped, sugar-coated breakfast cereal for children. The following (multiplicative exponential) demand function is being used: QD = 6,280P 2.1 5A 1.05 N 3.70 where QD = quantity demanded, in 10 oz: boxes P = price per box, in dollars A = advertising expenditures on daytime television, in dollars N = proportion of the population under 12 years old a. Determine the point price elasticity of demand for Tweetie Sweeties. b. Determine the advertising elasticity of demand. c. What interpretation would you give to the exponent of N
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The demand function for the breakfast cereals is estimated as follows:
img Here,
img is the quantity demanded (in 10 oz. boxes)
' P ' = Price per box (in $)
' A ' = Advertising expenditure on daytime television (in $)
' N ' = Proportion of the population under 12 years old
The point elasticity of demand ' e D ' measures the degree of responsiveness of a change in the quantity demanded for a given change in the market price level.
The price elasticity of demand is calculated as follows:
img (a)
The point-price elasticity of demand is calculated as follows:
img img Thus, the point price elasticity of demand is -2.15.
(b)
The advertising elasticity of demand is calculated as follows:
img img Thus, the point price elasticity of demand is +1.05.
(c)
The exponent of variable ' N ' indicates that one percentage change in the proportion of the population under 12 years old would bring about '3.70' percentage points change in demand for cereals.
It is the elasticity of demand with respect to proportion of the population under 12 years old.
The other variables ' P ' and ' A ' indicates that one percentage change in price level and advertising expenditure would bring about a negative '2.15' and a positive '1.05' percentage points change in the demand for cereals respectively.

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Cascade Pharmaceuticals Company developed the following regression model, using time-series data from the past 33 quarters, for one of its non-prescription cold remedies: Y = 1.04 + 0.24X1 0.27X2 whereY = quarterly sales ðin thousands of casesÞ of the cold remedy X1 = Cascade's quarterly advertising ð× $1,000Þ for the cold remedy X2 = competitors' advertising for similar products ð× $10,000) Here is additional information concerning the regression model: sb1 = 0:032 sb2 = 0:070 R 2 = 0:64 se = 1:63 F-statistic = 31:402 Durbin-Watson (d) statistic = 0.4995 a. Which of the independent variables (if any) appears to be statistically significant (at the 0.05 level) in explaining sales of the cold remedy b. What proportion of the total variation in sales is explained by the regression equation c. Perform an F-test (at the 0.05 level) of the overall explanatory power of the model. d. What additional statistical information (if any) would you find useful in the evaluation of this model
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The regression model developed by Cascade Pharmaceuticals Company, using time-series data from the past 33 quarters
img is
img where,
img are quarterly sales (1,000's of cases) of the cold remedy,
img Cascade's quarterly advertising
img for the cold remedy and
img is the competitors' advertising for similar products
img .
• The standard deviation for Cascade's quarterly advertising is
img .
• The standard deviation for competitors' advertising is
img .
• The standard error is
img .
• The coefficient of determination is
img and
img .
• Durbin-Watson statistic is
img .
The coefficient of Cascade's quarterly advertising is
img while that of the competitors' advertising is
img .
a) Perform the following t test to determine the statistical significance Cascade's quarterly advertising.
The null hypothesis is
img (no relationship between sales and Cascade's quarterly advertising expenditure)
And, the alternative hypothesis is
img (relationship exists between sales and Cascade's quarterly advertising expenditure)
At 5 percent of significance level, t-statistic for Cascade's quarterly advertising can be calculated as
img With 33 observations and 2 independent variables, a t -distribution for the sample statistic
img will have
img degrees of freedom.
From Table 2 of Appendix B, the t-value is obtained as
img Since the calculated value of t from is greater than
img , we reject null hypothesis
img .
Thus, we conclude that a linear and positive relationship exists between sales and Cascade's quarterly advertising expenditure at the 5 percent of significance level.
Similarly, perform the following t test to determine the statistical significance of competitor's quarterly advertising.
The null hypothesis is
img (no relationship between sales and competitor's quarterly advertising expenditure)
And, the alternative hypothesis is
img (relationship exists between sales and competitor's quarterly advertising expenditure)
At 5 percent of significance level, t-statistic for competitor's quarterly advertising can be calculated as
img With 33 observations and 2 independent variables, a t -distribution for the sample statistic
img will have
img degrees of freedom.
From Table 2 of Appendix B, the t-value is obtained as
img Since the calculated value of t from is less than
img , we reject null hypothesis
img .
Thus, we conclude that a linear and negative relationship exists between sales and competitor's quarterly advertising expenditure at the 5 percent of significance level.
Both independent variables are statistically significant in explaining the quarterly sales of cold remedy.
b) The coefficient of determination is
img . It implies 64 percent of the total variation is explained by the regression equation.
c) Perform the following F test to determine the overall explanatory power of the regression model.
The null hypothesis is
img (no relationship between sales and independent variables)
And, the alternative hypothesis is
img (relationship exists between sales and at least one of the independent variables)
With 33 observations and 2 independent variable, a F -distribution for each level of statistical significance has
img degrees of freedom for numerator and
img degrees of freedom for denominator.
( n is the total number of observations and k is the number of estimated regression parameters, including the constant term)
From Table 3 of Appendix B, the F -ratio for the above mentioned degrees of freedom at the 5 percent of significance level is 4.17.
The F statistic, however, in the regression model is calculated as 31.402. This value is significantly greater than the critical value.
Thus, we reject the null hypothesis of no relationship between sales and independent variables.
d) Durbin-Watson technique is used to check for autocorrelation in the regression model.
Autocorrelation is an economic problem that characterizes the existence of correlated successive error items in a time-series in a linear regression model.
The Durbin-Watson statistic is
img . Since
img (value implying no autocorrelation), there exists serial positive autocorrelation. As a consequence, standard error may be inflated.
With the presence of autocorrelation, the t-statistic used to test hypotheses about may yield misleading conclusions about the significance of independent variables.
Moreover, the
img and F tests are invalid under autocorrelation.

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Now omit both price and temperature from the regression equation. Should a marketing plan for soft drinks be designed that relocates most canned drink machines into low-income neighborhoods Why or why not TABLE 1 Soft Drink Demand Data (available as an Excel file on this book's Web site) img
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The Pilot Pen Company has decided to use 15 test markets to examine the sensitivity of demand for its new product to various prices, as shown in the following table. Advertising effort was identical in each market. Each market had approximately the same level of business activity and population. a. Using a linear regression model, estimate the demand function for Pilot's new pen. b. Evaluate this model by computing the coefficient of determination and by performing a t-test of the significance of the price variable. c. What is the price elasticity of demand at a price of 50 cents img
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Consider again the Sherwin-Williams Company example discussed in this chapter (see Table). Suppose one is interested in developing a multiple regression model with paint sales (Y) as the dependent variable and promotional expenditures (A) and selling price (P) as the independent variables. a. Determine the estimated regression line. b. Give an economic interpretation of the estimated slope (bs) coefficients. c. Test the hypothesis (at the 5 percent level of significance) that there is no relationship between the dependent variable and each of the independent variables. d. Determine the coefficient of determination. e. Perform an analysis of variance on the regression, including an F-test of the overall significance of the results (at the 5 percent level). f. Based on the regression model, determine the best estimate of paint sales in a sales region where promotional expenditures are $80(000) and the selling price is $12.50. g. Determine the point promotional and price elasticities at the values of promotional expenditures and selling price given in part (f). TABLE SHERWIN-WILLIAMS COMPANY DATA img
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Suppose an appliance manufacturer is doing a regression analysis, using quarterly time-series data, of the factors affecting its sales of appliances. A regression equation was estimated between appliance sales (in dollars) as the dependent variable and disposable personal income and new housing starts as the independent variables. The statistical tests of the model showed large t-values for both independent variables, along with a high r2 value. However, analysis of the residuals indicated that substantial autocorrelation was present. a. What are some of the possible causes of this autocorrelation b. How does this autocorrelation affect the conclusions concerning the significance of the individual explanatory variables and the overall explanatory power of the regression model c. Given that a person uses the model for forecasting future appliance sales, how does this autocorrelation affect the accuracy of these forecasts d. What techniques might be used to remove this autocorrelation from the model
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In a study of housing demand, the county assessor is interested in developing a regression model to estimate the market value (i.e., selling price) of residential property within his jurisdiction. The assessor feels that the most important variable affecting selling price (measured in thousands of dollars) is the size of house (measured in hundreds of square feet). He randomly selected 15 houses and measured both the selling price and size, as shown in the following table. img a. Plot the data. b. Determine the estimated regression line. Give an economic interpretation of the estimated slope (b) coefficient. c. Determine if size is a statistically significant variable in estimating selling price. d. Calculate the coefficient of determination. e. Perform an F-test of the overall significance of the results. f. Construct an approximate 95 percent prediction interval for the selling price of a house having an area (size) of 15 (hundred) square feet.
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The county assessor (see Exercise 9 of Chapter 4) is concerned about possible multicollinearity between the size (X 1 ) and total number of rooms (X 2 ) variables. Calculate the correlation coefficient between these two variables and diagnose the magnitude of the collinearity problem. Exercise 9 The county assessor (see Exercise 4) feels that the use of more independent variables in the regression equation might improve the overall explanatory power of the model. In addition to size, the assessor feels that the total number of rooms, age, and whether or not the house has an attached garage might be important variables affecting selling price. The data for the 15 randomly selected dwellings are shown in the following table. a. Using a computer regression program, determine the estimated regression equation with the four explanatory variables shown in the following table. b. Give an economic interpretation of each of the estimated regression coefficients. c. Which of the independent variables (if any) is statistically significant (at the 0.05 level) in explaining selling price d. What proportion of the total variation in selling price is explained by the regression model e. Perform an F-test (at the 5 percent significance level) of the overall explanatory power of the model. f. Construct an approximate 95 percent prediction interval for the selling price of a 15-year-old house having 1,800 sq. ft., 7 rooms, and an attached garage. img Exercise 4 Cascade Pharmaceuticals Company developed the following regression model, using time-series data from the past 33 quarters, for one of its nonprescription cold remedies: Y = -1:04 + 0:24X 1 - 0:27X 2 where Y = quarterly sales (in thousands of cases) of the cold remedy X 1 = Cascade's quarterly advertising (× $1,000) for the cold remedy X 2 = competitors' advertising for similar products (× $10,000) Here is additional information concerning the regression model: s b1 = 0:032 s b2 = 0:070 R 2 = 0:64 s e = 1:63 F-statistic = 31:402 Durbin-Watson (d) statistic = 0.4995 a. Which of the independent variables (if any) appears to be statistically significant (at the 0.05 level) in explaining sales of the cold remedy b. What proportion of the total variation in sales is explained by the regression equation c. Perform an F-test (at the 0.05 level) of the overall explanatory power of the model. d. What additional statistical information (if any) would you find useful in the evaluation of this model
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Omit price from the regression equation and observe the bias introduced into the
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Using the data in Table for the Sherwin-Williams Company, estimate a multiplicative exponential demand model (see Equation) for paint sales. TABLE SOFT DRINK DEMAND DATA (AVAILABLE AS AN EXCEL FILE ON THIS BOOK'S WEB SITE) img img b. Compare the results in part (a) (i.e., parameter estimates, standard errors, statistical significance) with the linear model developed in the chapter.
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Consider the Sherwin-Williams Company example discussed in this chapter (see Table). Suppose one is interested in developing a simple regression model with paint sales (Y) as the dependent variable and selling price (P) as the independent variable. a. Determine the estimated regression line. b. Give an economic interpretation of the estimated intercept (a) and slope (b) coefficients. c. Test the hypothesis (at the.05 level of significance) that there is no relationship (that is, = 0) between the variables. d. Calculate the coefficient of determination. e. Perform an analysis of variance on the regression, including an F-test of the overall significance of the results (at the.05 level). f. Based on the regression model, determine the best estimate of paint sales in a sales region where the selling price is $14.50. Construct an approximate 95 percent prediction interval. g. Determine the price elasticity of demand at a selling price of $14.50. TABLE SHERWIN-WILLIAMS COMPANY DATA img
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Interpret the coefficients and calculate the price elasticity of soft drink demand.
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Estimate the demand for soft drinks using a multiple regression program available on your computer.
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An estimate of the demand function for household furniture produced the following results: F = 0.0036Y 1.08 R 0.16 P 0.48 r 2 = 0.996 where F = furniture expenditures per household Y = disposable personal income per household R = value of private residential construction per household P = ratio of the furniture price index to the consumer price index a. Determine the point price and income elasticities for household furniture. b. What interpretation would you give to the exponent for R Why do you suppose R was included in the equation as a variable c. If you were a supplier to the furniture manufacturer, would you have preferred to see the analysis performed in physical sales units rather than dollars of revenue How would this change alter the interpretation of the price coefficient, presently estimated as 0.48
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The demand for haddock has been estimated as log Q = a + b log P + c log I + d log Pm where Q = quantity of haddock sold in New England P = price per pound of haddock I = a measure of personal income in the New England region Pm = an index of the price of meat and poultry If b = 2.174, c = 0.461, and d = 1.909, a. Determine the price elasticity of demand. b. Determine the income elasticity of demand. c. Determine the cross price elasticity of demand. d. How would you characterize the demand for haddock e. Suppose disposable income is expected to increase by 5 percent next year. Assuming all other factors remain constant, forecast the percentage change in the quantity of haddock demanded next year.
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A product manager has been reviewing selling expenses (i.e., advertising, sales commissions, etc.) associated with marketing a line of household cleaning products. The manager suspects that there may be some sort of diminishing marginal returns relationship between selling expenses and the resulting sales generated by these expenditures. After examining the selling expense and sales data for various regions (all regions are similar in sales potential) shown in the following table, however, the manager is uncertain about the nature of the relationship. img a. Using the linear regression model Y = + X where Y is sales and X is selling expenses, estimate , , and the r2 statistic by the least-squares technique. b. Using the exponential function model Y = X apply the double-logarithmic transformation to obtain a linear relationship that can be estimated by the least-squares technique. c. Applying the least-squares technique, estimate , , and the r2 statistic for the transformed (linear) model in part (b). (Note that the logarithms of the X and Y variables needed in the calculations are given in the table.) d. Based on the r2 statistics calculated in parts (a) and (c), which model appears to give a better fit of the data e. What implications does the result in part (d) have for the possible existence of a diminishing marginal returns relationship between sales and selling expenses as suggested by the manager f. What other transformations of the variables might we try to give a better fit to the data
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The following table presents data on sales (S), advertising (A), and price (P): img a. Estimate the following demand models: (i) S = + 1A þ 2 P (ii) S = A 1 P 2 b. Determine whether the estimated values of 1 and 2 are statistically significant (at the 0.05 level). c. Based on the value of R 2 and the F-ratio, which model gives the best fit
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