CHAPTER 4                                                           DEMAND ESTIMATION 

1. Use the method of least squares to calculate the intercept and slope of the linear relationship between quantity demanded (Q) and price (P) from the data set that follows.

        Q         P

        2          6

        3          5

        5          3

        2          7

Solution:

Q       P        Q − ΣQ/n       P − ΣP/n       (Q − ΣQ/n)(P − ΣP/n)       (P − ΣP/n)²

 2       6           −1                  0.75                       −0.75                         0.5625

 3       5             0                −0.25                          0.0                           0.0625

 5       3             2                −2.25                       −4.5                            5.0625

 2       7           −1                   1.75                       −1.75                          3.0625

ΣQ = 12        ΣP = 21              ΣQ/n = 12/4 = 3                              ΣP/n = 21/4 = 5.25

Σ(Q − ΣQ/n)(P − ΣP/n) = −7         Σ(P − ΣP/n)² = 8.75            b = −7/8.75 = −0.80

a = 3 − (−0.8)(5.25) = 7.2       Equation: Q = 7.2 − 0.80P

2. Use the method of least squares to calculate the intercept and slope of the linear relationship between quantity demanded (Q) and advertising (A) from the data set that follows.

Q   A

2    0

3    1

5    4

10  7

43  1

Solution:

Q        A        Q − ΣQ/n     A − ΣA/n      (Q − ΣQ/n)(A − ΣA/n)       (A − ΣA/n)²

2         0            −3                −3                            9                                    9

3         1            −2                 −2                            4                                    4

5         4              0                   1                             0                                    1

10       7              5                   4                           20                                  16

ΣQ = 20            ΣA = 12          ΣQ/n = 20/4 = 5           ΣA/n = 12/4 = 3

Σ(Q − ΣQ/n)(A − ΣA/n) = 33         Σ(A − ΣA/n)2 = 30               b = 33/30 = 1.1

a = 5 − (1.1)(3) = 1.7                           Equation: Q = 1.7 − 1.1A

3. Calculate the equation of the linear function that is plotted on the graph.

Solution:

Slope = (8 − 5)/(5 − 9) = −3/4 = −0.75

Intercept = 8 − (−0.75)(5) = 11.75

Equation: Y = 11.75 − 0.75X

4. Calculate the equation of the linear function that is plotted on the graph.

Solution:

Slope = (9 − 6)/(10 − 5) = 3/5 = 0.60

Intercept = 9 − (0.60)(10) = 3.0

Equation: Y = 3.0 + 0.60X

5. Just the Fax, Inc. (JTF) has hired you as a consultant to analyze the demand for its line of telecommunications devices in 35 different market areas. The available data set includes observations on the number of thousands of units sold by JTF per month (Q_X), the price per unit charged by JTF (PX), the average unit price of competing brands (P_Z), monthly advertising expenditures by JTF (A), and average gross sales (in thousands of dollars) of businesses in the market area (I). The results of a regression analysis (with standard

errors in parenthesis) are given below.

QX = 300      − 6PX      + 2PZ       + 0.04A      + 0.01I                                           

        (200)        (1.8)       (0.8)         (0.03)      (0.004)

R² = 0.91             S.E.E. = 3.6

(i) Evaluate the statistical significance of the equation as a whole and of each of its coefficients.

(ii) The average values of the independent variables in the data set used to estimate the equation are P_X = $195, P_Z = $225, A = $11,000, and I = $200,000. Calculate a point estimate of JTF's average sales and a 95 percent interval estimate of sales based on these values.

Solution:

(i) The F test statistic is calculated from R² as follows:

F = (0.91/4)/(0.09/30) = 75.83 (highly significant)

The t ratio for the constant term is t = 300/200 = 1.5 (not significant)

The t ratios for each of the independent variables are as follows:

P_X: t = −6/1.8 = −3.33 (highly significant)

P_Z: t = 2/0.8 = 2.5 (highly significant)

A: t = 0.04/0.03 = 1.33 (not significant)

I: t = 0.01/0.004 = 2.5 (highly significant)

(ii) The point estimate of Q_X = 22. The interval estimate is approximately 14.8 to

29.2, using a Z value of 2.

6. The Drag Net Fishing Company (DNF) has hired you as a consultant to analyze the demand by local restaurants for fresh fish. The available data set includes monthly observations collected over the past 5 years on the number of hundreds of pounds of fish purchased by local restaurants per month (Q), the price per pound of fish (P), the average cost of a meal at a local restaurant (M), a seasonal variable (S) that is equal to one during the tourist season and zero otherwise, and average household income (in thousands of dollars) in the area. The results of a regression analysis (with standard errors in parenthesis) are given below.

Q =          110         − 0.2P          − 0.6M          + 4.2S         + 0.05I

                 (42)          (0.12)           (0.28)           (0.70)        (0.024)

R² = 0.74                                              S.E.E. = 12.9

(i) Evaluate the statistical significance of the equation as a whole and of each of its coefficients.

(ii) The average values of independent variables in the data set that was used to estimate the equation are P = $8, M = $14, and I = $40,000. Calculate a point

estimate of the restaurant demand for fish and a 95 percent interval estimate when it is tourist season. Also, calculate a point estimate of the restaurant demand for fish and a 95 percent interval estimate when it is not tourist season.

Solution:

(i) The F test statistic is calculated from R² as follows:

F = (0.74/4)/(0.26/55) = 39.13 (highly significant)

The t ratio for the constant term is t = 110/42 = 2.6 (highly significant)

The t ratios for each of the independent variables are as follows:

P: t = −0.2/0.12 = −1.67 (barely significant)

M: t = −0.6/0.28 = −2.14 (significant)

S: t = 4.2/0.70 = 6.00 (highly significant)

I: t = 0.05/0.024 = 2.08 (significant)

(ii) During the tourist season, the point estimate of Q = 144. The interval estimate is approximately 118.2 to 169.8, using a Z value of 2. During the off season, the point estimate of Q = 102. The interval estimate is approximately 76.2 to 127.8.

7. Florid Technologies is a manufacturer of exercise machines. Their best-selling device, a mechanical swimming simulator, has been on the market for several years. The demand function for the simulator was estimated in log-linear form using time-series data. The results are presented below.

Q_X = 110P_X^−0.20 P_Y^0.30 P_Z^−0.10 A^0.10 I^0.01

The number of simulators sold per week (Q_X) was found to depend on the price charged for a simulator (P_X), the average monthly cost of membership at a health club (P_Y), the cost of an accessory package designed for use with the simulator (PZ), monthly advertising expenditures (A) in thousands, and average annual household income (I) in thousands.

(i) If P_X = $595, P_Y = $45, P_Z = $99.85, A = $11,000, and I = $44,000, how many

simulators can Florid Technologies expect to sell in a week?

(ii) Interpret the price elasticity, cross-price elasticities, advertising elasticity, and income elasticity of demand for simulators.

(iii) The president of Florid Technologies plans to increase the simulator's price by 10 percent and to increase advertising expenditures by 5 percent. By what percentage can sales of simulators be expected to change? Will total revenue increase, decrease, or remain the same? Explain your answer.

(iv) If the price of a simulator is increased by 10 percent and the cost of an accessory package is reduced by 20 percent, what effect will this have on simulator sales?

Solution:

(i) There are two ways to solve this problem. The first is by substituting into the

power function directly. This yields:

Q_X = (110)(595)^−0.20(45)^0.30(99.85)^−0.10(11)^0.10(44)^0.01 = 80

The second way is to convert to log-linear form and then to take the antilog of the solution. This yields:

lnQ_X = ln (110) − 0.20 ln (595) + 0.30 ln (45) − 0.10 ln (99.85)

+ 0.10 ln (11) + 0.01 ln (44) = 4.3828

lnQ_X = 4.70 − (0.20)(6.39) + (0.30)(3.81) − (0.10)(4.60) + (0.10)(2.40)

+ (0.01)(3.78) = 4.3828

Q_X = e^4.3828 = 80

(ii) The price elasticity of demand is −0.20, which indicates that demand for

simulators is inelastic. The cross-price elasticity of demand with health club

membership is positive, which indicates that these two goods are substitutes. The cross-price elasticity of demand with simulator accessories is negative, which indicates that these two goods are complements. The advertising elasticity is positive, which indicates that an increase in advertising will increase demand. The income elasticity of demand is positive and close to zero, which indicates that simulators are normal goods but that demand is largely unresponsive to changes in average income.

(iii) The demand for simulators will decrease by 1.5 percent:

(10 percent)(−0.20) + (5 percent)(0.10) = −1.5 percent

The percentage increase in price exceeds the percentage decrease in quantity sold, so the effect of the changes will be to increase total revenue.

(iv) The demand for simulators will not change:

(10 percent)(−0.20) + (−20 percent)(−0.10) = 0

8. The log-linear demand function for Beckler's Frozen Pizzas is:

lnQ_X = 4 − 3.80 lnP_X + 0.30 lnP_Y + 0.15 lnS + lnA + 1.50 lnI

The number of pizzas sold per week (Q_X) depends on the price charged for a pizza (P_X), the price charged for a competitor's brand of pizza (PY), the percentage of single-parent families (S), monthly advertising expenditures (A) in thousands, and average annual household income (I) in thousands.

(i) If P_X = $4.00, P_Y = $3.50, S = 40%, A = $5,000, and I = $40,000, how many

pizzas can Beckler's expect to sell in a week?

(ii) Interpret the price elasticity, cross-price elasticity, family structure elasticity,

advertising elasticity, and income elasticity of demand for pizzas.

(iii) The president of Beckler's plans to increase the price of their pizzas by 25 percent and to increase advertising expenditures by 10 percent. By what percentage can the number of pizzas sold by Beckler's be expected to change? Will total revenue increase, decrease, or remain the same? Explain your answer.

(iv) If Beckler's lowers the price of its pizzas by 5 percent and increases advertising expenditures by 10 percent while Beckler's competitor lowers the price of its pizza by 20 percent, what effect will this have on the number of Beckler's pizzas sold per week?

Solution:

(i) lnQ_X = 4 − 3.80 ln(4) + 0.30 ln(3.5) + 0.15 ln(40) + ln(5) + 1.50 ln(40)

lnQ_X = 4 − (3.80)(1.39) + (0.30)(1.25) + (0.15)(3.69) + 1.61 + (1.50)(3.69)

lnQ_X = 6.7915 (using rounded values, otherwise 6.8113)

Q_X = 890 if lnQ_X = 6.7915 or Q_X = 908 if lnQ_X = 6.8113

(ii) The price elasticity of demand is −3.80, which indicates that demand for Beckler's pizza is elastic. The cross-price elasticity of demand with the competitor's brand of pizza is positive, which indicates that these two goods are substitutes. The family structure elasticity is positive, which indicates that pizza demand will increase if the proportion of single-parent families increases. The advertising elasticity is equal to 1, which indicates that a 1 percent increase in advertising will increase demand by 1 percent. The income elasticity of demand is positive and greater than zero, which indicates that Beckler's pizzas are normal goods.

(iii) The demand for Beckler's pizzas will decrease by 85 percent:

(25 percent)(−3.80) + (10 percent)(1.00) = 85 percent

The percentage decrease in quantity sold far exceeds the percentage increase in price, so the effect of the changes will be to decrease total revenue.

(iv) The demand for Beckler's pizzas will increase by 23 percent:

(5 percent)(−3.80) + (10 percent)(1.00) + (−20 percent)(0.30) = 23 percent