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Chapter 4

Estimating Demand

Solutions to Exercises

1. a. The coefficient -0.582 is the price elasticity of demand. The coefficient 0.401 is the income elasticity of demand. The coefficient -0.211 is the cross price elasticity of demand between gasoline and cars, complementary goods. b. The t value for the "own" price coefficient is 5.54, which is significant at the less than 1% level. The t value for income elasticity is 2.06, which is significant at better than the 5% level. The cross price elasticity t-value is 1.35, which is significant only at the 20% level. 2. a. -2.174 b. 0.461 c. 1.909 d. The demand is very price elastic and heavily influenced by the price of competitive goodsmeat and poultry. e. The quantity of haddock demanded would increase by .461(5%) = 2.31% 3. a. Dependent variable: Sales Variable Parameter Standard T-ratio Estimate Error _______________________________________________________________________ Intercept 1 2.4365 1.105 2.205 Price 1 -.3750 .239 -1.570 Income 1 5.9492 2.140 2.780 Advertising 1 6.2500 1.950 3.205 DF

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Source of Variation Regression Residual Total R-Square: 0.743 F-Ratio: 21.250 b. H0: ßj = 0; Ha: ßj 0

Chapter 4/Estimating Demand

DF 3 22 25 Sum of Squares 1187.343 409.750 1597.093 Mean Squares 395.781 18.625

Reject H0 at the .05 level of significance if t (i.e., T-ratio) < -t.025,22 = -2.074 or t > t.025,22 = +2.074. Since the calculated t-value is greater than the t-value from the table for the income and advertising variables, one rejects the hypothesis at the .05 significance level that there is no relationship between these variables and sales. c. H0: 1 = 2 = 3 = 0 Ha : At least one j 0 Reject H0 at the .05 level of significance if F (i.e., F-Ratio) > F.05,3,22 = 3.05. Since the calculated F-value is greater than the F-value from the table, one rejects at the .05 significance level the hypothesis that there is no relationship between any of the explanatory variables and sales. 4. a. Dependent variable: SALES N: 10 Multiple R: 0.874 Squared multiple R: 0.764 Adjusted squared multiple R: 0.697 Standard error of the estimate: 17.062 Variable Coefficient Std error Std coef CONSTANT 344.585 84.245 0.000 PROMEXP 0.106 0.164 0.177 SELLPR -12.112 4.487 -0.736 Tolerance 0.4531851 0.4531851 T P(2 tail) 4.090 0.005 0.648 0.537 -2.699 0.031

Source Regression Residual

Analysis of Variance Sum-of-squares DF Mean-square 6612.203 2 3306.102 2037.797 7 291.114

F-ratio 11.357

P 0.006

Y = 344.585 + .106X1 - 12.112X2

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Chapter 4/Estimating of Demand

b. b1 = .106

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A one-unit (i.e., $1,000) increase in promotional expenditures increases expected sales by .106 (× 1,000) = 106 gallons, all other things being equal. A one-unit (i.e., $1.00) increase in the selling price decreases expected sales by 12.112 (× 1,000) = 12,112 gallons, all other things being equal. c. The computer output indicates that only selling price (X2) is statistically significant at the 0.05 level. The t-value for significance is t.025,7 = 2.365. d. R2 = 0.764 The model explains about 76 percent of the variation in paint sales.

e. The F-ratio from the computer output is 11.357. The F-value for statistical significance at the .05 level is F.05,2,7 = 4.74. Therefore we reject the null hypothesis and conclude that the independent variables are useful in explaining paint sales. f. Y' = 344.585 + 0.106(80) - 12.112(12.50) = 201.665 or 201,665 gallons. g. (i) EA= (Y/X1)(X1/Y) = .106 (80/201.665) = .0420 (ii) EL = (Y/X2)(X2/Y) = -12.112 (12.50/201.665) = -.751 5. a. Variable X1 X2 Coefficient .24 -.27 Standard Error t-statistic Decision* .032 7.50 Significant .070 -3.86 Significant

*Variable Xi is significant at the .05 level if t > t.025,30 = +2.042 or t < -2.042. b. R2 = 0.64 Approximately 64% of the variation in sales is "explained" by the regression equation.

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Chapter 4/Estimating Demand

c. Ho: All ßj = 0; Ha: At least one ßj 0. Reject Ho at the .05 level if the calculated F-value is greater than F.05,2,30 = 3.32. Since F = 31.402 > 3.32, one rejects Ho and concludes that the regression model "explains" a significant proportion of the variation in quarterly sales of the cold remedy. d. Ho: No positive or negative autocorrelation Ha: Positive or negative autocorrelation Reject Ho at the .05 significance level if the calculated Durbin-Watson d-statistic is either less than dL = 1.16 from Table 6 (m = 3 and n = 33) or greater than (4 - dU) = (4 -1.55) = 2.45. Since d = .4995, one concludes that significant (positive) autocorrelation is present. e. The presence of autocorrelation will cause the least-squares procedure to underestimate the sampling variance of the estimated regression coefficients. As a result the t-test will no longer yield reliable conclusions concerning the statistical significance of the individual explanatory variables. Also, the overall measures of goodness-of-fit and explanatory power of the regression equation, namely the coefficient of determination and F-test, will no longer provide a reliable indication of the significance of the economic relationship obtained. The correlation coefficient between X1 and X2 would be useful in checking for multicollinearity.

6. a. ED = -.48 (exponent of P) EY = 1.08 (exponent of Y) b. The percentage change in furniture expenditures resulting from a one-percent change in the value of private residential construction per household. c. Depends on how results are to be used: (1) physical unitsproduction planning (2) actual dollar salesfinancial planning If F is expressed in constant dollar terms, then Y should be also.

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Chapter 4/Estimating of Demand

7. a.

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b. Y = 6.279 + 3.956X The estimated slope coefficient indicates that the selling price increases by $3,956 for each 100 square feet increase in the size of a house. c. se = 10.132 sb = .396 t = (3.956 - 0)/.396 = 9.990 Since the calculated t-value is greater than the t-value from the table (t.025,13 = -2.160 or + 2.160), one rejects the hypothesis at the .05 significance level that there is no relationship between the selling price and the size of a house. d. R2 = .885 e. Source of Variation Regression Residual Total Sum of Squares 10,244 1,335 11,579 Degrees of Freedom 1 13 14 Mean Squares 10,244 102.7

F = MSR/MSE = 10,244/102.7 = 99.75

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Chapter 4/Estimating Demand

Since the calculated F-value is greater than the F-value from the table (F.05,1,13 = 4.67), one rejects at the .05 significance level the hypothesis that there is no relationship between the selling price and the size of a house. f. y' = 6.279 + 3.956(15) = 65.619 or $65,619

Approximate 95% prediction interval: y' = 65.619 ± 2(10.132) = 45.355 to 85.883, or $45,355 to $85,883. 8. a. -0.95 = price elasticity of demand 1.40 = income elasticity of demand 0.3 = advertising elasticity of demand 0.2 = cross price elasticity of demand 0.6 = population elasticity of demand b. Yes, all of the signs are correct. Demand for gasoline appears to be slightly price inelastic. c. Quantity demanded would appear to fall to zero. This highlights the importance of using demand functions to make inferences within a relevant (and reasonable) range of values. 9. a. Y = 28.915 - 19.105X (-6.095) (Price input in decimal form) (t statistic in parentheses) b. R2 = .74 (Model explains 74% of variation in quantity demanded.)

Ho: ß = 0; Ha: ß 0; Reject Ho at 0.05 significance level if t>t.025,13 = +2.160 or t<-2.160. Since t = -6.095<-2.160, reject Ho (i.e., price is statistically significant in explaining demand).

c. Y' = 28.915 -19.105(.50) = 19.363 (thousand pens) ED = -19.105(.50/19.363) = -0.49 10. a. Dependent variable: Price Source Model Error C Total DF 4 10 14 Sum of Squares 10339.040 1239.798 11578.837 Mean Square 2584.760 123.980 F Value 20.848 Prob > F 0.0001

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Chapter 4/Estimating of Demand

Root MSE Dep Mean C.V. Variable Intercep Size Rooms Age Garage 11.134621 100.587 11.06968 DF 1 1 1 1 1 Parameter Est. -14.735136 3.921432 3.585118 -0.118145 -2.831695 R-Square ADJ R-Sq 0.8929 0.8501

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Std. Error 27.483494 0.755262 4.470708 0.640698 9.716504

t-ratio -0.536 5.192 0.802 -0.184 -0.291

Prob > |t| 0.6036 .0004 0.4412 0.8574 0.7767

Y' = -14.7351 + 3.9214X1 + 3.5851X2 - 0.1181X3 - 2.8317X4 b. a = -14.7351 No significant economic meaning. b1 = 3.9214 An increase of 100 sq. ft. in size increases the expected selling price by $3921, all other things remaining constant. b2 = 3.5851 An increase of 1 room increases the expected selling price by $3,585, all other things remaining constant. b3 = -.1181 An increase of 1 year in age decreases the expected selling price by $118, all other things remaining constant. b4 = -2.8317 An attached garage decreases the expected selling price by $2,832, all other things remaining constant. c. From the computer output, only X1(size) is significant in explaining the selling price at the 5% significance level or better. d. R-Square = 0.8929. The regression model explains about 89% of the variation in selling price. e. From the computer output, F is significant at the .0001 level.

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Chapter 4/Estimating Demand

f. y' = -14.7351 + 3.9214(18) + 3.5851(7) - .1181(15) -2.8317(1) = 76.3426 se = 11.1346 (Root MSE from computer output) y' ± 2 se = 76.3426 ± 2(11.1346) = 54.0734 to 98.6118, or $54,073 to $98,612

11. a. Dependent variable (Y)Amount of Life Insurance Independent variable XAnnual Income b.

c. Y = 11.148 + 1.492X The estimated slope coefficient (b = 1.492) indicates that the amount of life insurance held by executives increases by 1.492 x $1000 = $1,492 for each $1000 increase in annual income. d. se = 43.34 sb = 0.565 t = (1.492 - 0)/.565 = 2.641 Since the calculated t-value is greater than the t-value from the table (t.025,10 = -2.228 or +2.228), one rejects the hypothesis at the .05 significance level that there is no relationship between the amount of life insurance held and annual income.

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Chapter 4/Estimating of Demand

e. R2 = .41 f. Source of Variation Regression Residual Total Sum of Squares 13,088 18,787 31,875 F.05,1,10 = 4.96 F = MSR/MSE = 13,087/1879 = 6.966 Degrees of Freedom 1 10 11 Mean Squares 13,088 1,879

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Since the calculated F-value is greater than the F-value from the table, one rejects at the .05 significance level the hypothesis that there is no relationship between the amount of life insurance held and annual income. g. y' = 11.148 + 1.492(80) = 130.508 or $130,508 Approximate 95% prediction interval: y' ± 2(43.344) = 130.508 ± 86,688 = 43.82 to 217.196, or $43,820 to $217,196. 12. a. ED = -2.15 (the exponent of P) b. EA = 1.05 (the exponent of A) c. The exponent of N (3.70) represents the elasticity of the quantity demanded with respect to the proportion of the population under12 years of age. It indicates that the quantity demanded will increase (decrease) by 3.70 percent for each one percent increase (decrease) in the proportion of the population under 12 years of age. 13. a. Y=390.376 -14.263X2 b. The estimated intercept value of 390.376 indicates that sales (Y) will be equal to 390.376 (X1000) = 390,376 gallons when the selling price (X2) is equal to zero. This value of X2 lies far outside the range over which the regression line was estimated (recall that the lowest selling price in the sample was $12.00 in sales region 9) and the sales estimate has no practical economic significance. The estimated slope coefficient of -14.263 indicates that expected sales (Y) will decrease by 14.263 (X1000) = 14,263 gallons for each additional $1.00 increase in the selling price (X2).

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c. H0: ß2=0

Chapter 4/Estimating Demand

Ha: ß2 0 b2= -14.263 sb2 = 2.909

t = (-14.263 - 0)/2.909 = -4.903 Since the calculated t-value is less than the t-value from the table (t.025,8 = -2.306), one rejects the null hypothesis at the .05 significance level that there is no relationship (i.e., 2=0) between paint sales and the selling price. d. R2 = .75 The regression equation, with selling price as the independent variable, explains 75 percent of the variation in paint sales in the sample. e. Source of Variation Regression Residual Total H0: 2=0 Sum of Squares 6489.812 2160.188 8650.000 Degrees of Freedom 1 8 9 Mean Squares 6489.812 270.024

Ha: 2 0 F = MSR/MSE = 6489.812/270.024 = 24.034 Since the calculated F-value is greater than the F-value from the table (F.05,1,8 = 5.32), one rejects the null hypothesis at the .05 significance level that there is no relationship between selling price And paint sales. f. Xp = $14.50 se = 16.432 Y' = 390.376 - 14.263(14.50) = 183.563 or 183,563 gallons Y' - 2se = 183.563 - 2(16.432) = 150.699 Y' + 2se = 183.563 + 2(16.432) = 216.427 or from 150,699 gallons to 216,427 gallons. g. ED = (dY/dX2)(X2/Y) = -14.263 (14.50/183.563) = -1.13

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Chapter 4/Estimating of Demand

14. a. QD = 10,425 - 2910PX + .028A + 11,100 Pop QD/P = -2910

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ED = -2910(5/29,425) = - 0.49

QD = 10,425 - 2910(5) + .028(1,000,000) + 11,100(.5) QD = 29,425

QD = 10,425 - 2910(10) + .028(1,000,000) + 11,100(.5) QD = 14,875 ED = -2910(10/14,875) = -1.96 b. QD/A = 0.028

QD = 10,425 - 2910(5) + .028(2,000,000) + 11,100(.5) QD = 57,425 EA = .028(2,000,000/57,425) = 0.98

c. Yes, all t values are well in excess of t.025,21 = 2.074. ______________________________________________________________________

Solution to Case Exercise: Soft Drinks

1. The linear demand estimation is as follows: R2 = 0.70 QD= 514.2 + 2.93 TEMP + 1.22 INCOME - 242.9 PRICE (4.12) (O.80) (-5.58) SSE = 38.26 and for the log-linear model, log QD= 1.050 + 1.12 log TEMP + 0.22 log INCOME - 3.12 log PRICE R2 = 0.67 (1.72) (4.23) (1.19) (- 4.92) SSE = 0.11 where the numbers in parentheses are t-scores. 2. Both temperature and price are statistically significant with expected signs while income is insignificant in its effect on soft drink demand. For the linear model, the price elasticity of demand is (Q/P) × (Mean P/Mean Q) -242.97 x ($2.2025/158.2083) = - 3.38 and for the log-linear model -3.12.

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Chapter 4/Estimating Demand

This point elasticity at the mean price and quantity across the states is in the elastic range, as expected. Note, however, that these are market-level price elasticities, so no firm behavior is directly implied by this estimate. Nevertheless, an elastic demand at the market level does imply elastic firm-level demand at comparable prices, comparable price sensitivity, and the smaller quantities facing each firm.

3. Omitting price from the regression, one obtains for the log-linear model log QD= - 0.16 + 1.72 log TEMP - 0.152 log INCOME (5.96) (- 0.73) 4. Omitting both price and temperature yields a linear model as follows: QD = 254.6 - 5.37 INCOME R2 = 0.11 (- 2.11) SSE = 64.2 For the log-linear model, one obtains QD = 4.47 - 0.552 INCOME (- 2.13)

R2 = 0.49 SSE = 0.137

R2 = 0.09 SSE = 0.18

No, a marketing plan should not be designed specifically to introduce canned soft drink machines into low-income neighborhoods. And students should not offer the negative and significant income parameter estimate above as their reason. The above regression does NOT call for relocating canned soft drink machines away from low-income neighborhoods. The regression coefficient on income has been biased downward by the omission of price and temperature enough to make an insignificant factor appear negative and significant in its effect on demand. This illustrates the critical importance of using analytical reasoning and demand theory to correctly specify a regression model.

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