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# Chapter 3 Answers to Questions and Problems

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Chapter 3 Answers to Questions and Problemsto,TO,and,AND

Chapter 3: Answers to Questions and Problems

1.

a. When P = \$12, R = (\$12)(1) = \$12. When P = \$10, R = (\$10)(2) = \$20. Thus,

the price decrease results in an \$8 increase in total revenue, so demand is elastic

over this range of prices.

b. When P = \$4, R = (\$4)(5) = \$20. When P = \$2, R = (\$2)(6) = \$12. Thus, the

price decrease results in an \$8 decrease total revenue, so demand is inelastic over

this range of prices.

c. Recall that total revenue is maximized at the point where demand is unitary

elastic. We also know that marginal revenue is zero at this point. For a linear

demand curve, marginal revenue lies halfway between the demand curve and the

vertical axis. In this case, marginal revenue is a line starting at a price of \$14 and

intersecting the quantity axis at a value of Q = 3.5. Thus, marginal revenue is 0 at

3.5 units, which corresponds to a price of \$7 as shown below.

Price\$14

\$12

\$10

\$8

\$6

\$4

\$2Demand

\$0

MR0123456Quantity

Figure 3-1

Managerial Economics and Business Strategy, 4e Page 1

2.

a. At the given prices, quantity demanded is 700 units:

d. Substituting the relevant information into Q，？？，10002154.02400700;；;；x

P154xthe elasticity formula gives: . Since this is less E，？，？，？220.44,QPxxQ700x

than one in absolute value, demand is inelastic at this price. If the firm charged a

lower price, total revenue would decrease.

b. At the given prices, quantity demanded is 300 units:

d. Substituting the relevant information into Q，？？，10002354.02400300;；;；x

~?P354~?xthe elasticity formula gives: . Since this is E，？，？，？222.36,QP????xxQ300?(x?(

greater than one in absolute value, demand is elastic at this price. If the firm

increased its price, total revenue would decrease.

c. At the given prices, quantity demanded is 700 units:

dQ，？？，10002154.02400700. Substituting the relevant information into ;；;；x

~?P400~?Zthe elasticity formula gives: . Since this E，，，.02.020.011,QP????xZQ700?(x?(

number is positive, goods X and Z are substitutes.

3.

a. The own price elasticity of demand is simply the coefficient of ln P, which is x

0.5. Since this number is less than one in absolute value, demand is inelastic. b. The cross-price elasticity of demand is simply the coefficient of ln P, which is y

2.5. Since this number is negative, goods X and Y are complements. c. The income elasticity of demand is simply the coefficient of ln M, which is 1.

Since this number is positive, good X is a normal good.

d. The advertising elasticity of demand is simply the coefficient of ln A, which is 2.

4.

d%Qx，？2a. Use the own price elasticity of demand formula to write . Solving, 5

we see that the quantity demanded of good X will decrease by 10 percent if the

price of good X increases by 5 percent.

d%Qx，？6b. Use the cross-price elasticity of demand formula to write . Solving, 10

we see that the demand for X will decrease by 60 percent if the price of good Y

increases by 10 percent.

d%Qx4c. Use the formula for the advertising elasticity of demand to write . 2

Solving, we see that the demand for good X will decrease by 8 percent if

Page 2 Michael R. Baye

d%Qx. Solving, we d. Use the income elasticity of demand formula to write 33

see that the quantity demanded of good X will decrease by 9 percent if income

decreases by 3 percent.

5055. Using the cross price elasticity formula, . Solving, we see that the price %Py

of good Y would have to decrease by 10 percent in order to increase the consumption

of good X by 50 percent.

6. Using the change in revenue formula for two products,

. Thus, a 1 percent increase in the ，(;；;；;；R\$30,00012.5\$70,0001.1.01\$320

price of good X would cause revenues from both goods to increase by \$320.

7. Table 3-1 contains the answers to the regression output.

SUMMARY OUTPUT

Regression Statistics

Multiple R0.62

R Square0.39

Standard Error190.90

Observations100.00

ANOVA

degrees of freedomSSMSFSignificance F

Regression2.002,223,017.771,111,508.8830.500.00

Residual97.003,535,019.4936,443.50

Total99.005,758,037.26

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%

Intercept187.15534.710.350.73-880.561,254.86

Price of X-4.320.696.260.00-5.69-2.96Table 3-1 Income0.090.024.470.000.050.14

da. QPM，？？187.154.32.09. xx

b. Only the coefficients for the Price of X and Income are statistically significant at

the 5 percent level or better.

c. The R-square is fairly low, indicating that the model explains only 39 percent of

the total variation in demand for X. The adjusted R-square is only marginally

lower (37 percent), suggesting that the R-square is not the result of an excessive

number of estimated coefficients relative to the sample size. The F-statistic,

however, suggests that the overall regression is statistically significant at better

than the 5 percent level.

ˆ8. The approximate 95 percent confidence interval for a is a2102. Thus, you ˆa

can be 95 percent confident that a is within the range of 8 and 12. The approximate

Managerial Economics and Business Strategy, 4e Page 3

ˆb22.51. Thus, you can be 95 95 percent confidence interval for b is ˆb

percent confident that b is within the range of 3.5 and 1.5.

9. The result is not surprising. Given the available information, the own price elasticity

137of demand for Palm’s brand of PDAs is . Since this number is E8.06Q,P17

greater than one in absolute value, demand is elastic. By the total revenue test, this means that a reduction in price will increase revenues.

10. The regression output is as follows:

SUMMARY OUTPUT

Regression Statistics

Multiple R0.97

R Square0.94

Standard Error0.00

Observations49

ANOVA

dfSSMSFSignificance F

Regression20.007020.004370.380.0000

Residual460.000440.000

Total480.00745

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%

Intercept1.290.413.120.000.462.12

LN Price-0.070.00-26.620.00-0.08-0.07

LN Income-0.030.09-0.330.74-0.220.16

Table 3-2

ln1.290.07ln0.03lnQPM，？？Thus, the demand for your batteries is given by .

Since this is a log-linear demand equation, the best estimate of the income elasticity of demand for your product is -.03: Your batteries are an inferior good. However, note the estimated income elasticity is very close to zero (implying that a 3 percent reduction in global incomes would increase the demand for your product by less than one tenth of one percent). More importantly, the estimated income elasticity is not statistically different from zero (the 95 percent confidence interval ranges from a low of -.22 to a high of .16, with a t-statistic that is well below 2 in absolute value). On balance, this means that a 3 percent decline in global incomes is unlikely to impact the sales of your product. Note that the R-square is reasonably high, suggesting the

model explains 94 percent of the total variation in the demand for this product. Likewise, the F-test indicates that the regression fit is highly significant.

Page 4 Michael R. Baye

11. Based on this information, the own price elasticity of demand for Big G cereal is

3. Thus, demand for Big G cereal is elastic (since this number is E1.5Q,P2

greater than one in absolute value). Since Lucky Charms is one particular brand of

cereal for which even more substitutes exist, you would expect the demand for Lucky

Charms to be even more elastic than the demand for Big G cereal. Thus, since the

demand for Lucky Charms is elastic, one would predict that the increase in price of

Lucky Charms resulted in a reduction in revenues on sales of Lucky Charms.

d%Q1.7512. Use the income elasticity formula to write . Solving, we see that coffee 4

purchases are expected to decrease by 7 percent.

13. To maximize revenue, GM should charge the price that makes demand unit elastic.

Using the own price elasticity of demand formula,

P~?. Solving this equation for P implies that the 1.251E，？，？;；QP,??100,0001.25P?(

P\$40,000revenue maximizing price is .

14. Using the change in revenue formula for two products,

, so revenues will increase ，(;；;；;；R\$60012.5\$4000.2.01\$9.8 million

by \$9.8 million.

15. The estimated demand function for residential heating fuel is

d, where is the price Q136.9691.69P43.88P11.92P0.05MPRHFRHFNGERHF

of residential heating fuel, is the price of natural gas, is the price of electricity, PPNGE

and M is income. However, notice that coefficients of income and the price of

electricity are not statistically different from zero. Among other things, this means

that the proposal to increase the price of electricity by \$5 is unlikely to have a

statistically significant impact on the demand for residential heating fuel. Since the

coefficient of is -91.69, a \$2 increase in would lead to 183.38 unit PPRHFRHF

reduction in the consumption of residential heating fuel (since (-91.69)(\$2) = - 183.38

units). Since the coefficient of is 43.88, a \$1 reduction in would lead to PPNGNG

43.88 unit reduction in the consumption of residential heating fuel (since (43.88)(-\$1)

= -43.88). Thus, the proposal to increase the price of residential heating fuel by \$2

would lead to the greatest expected reduction in the consumption of residential

heating fuel.

Managerial Economics and Business Strategy, 4e Page 5

16. The regression output is as follows:

SUMMARY OUTPUT

Regression Statistics

Multiple R0.97

R Square0.94

Standard Error0.06

Observations41

ANOVA

dfSSMSFSignificance F

Regression12.242.24599.260.00

Residual390.150.00

Total402.38

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%

Intercept4.290.1237.170.004.064.53

ln (Price)-1.380.06-24.480.00-1.50-1.27

Table 3-3

ln4.291.38lnQP，？Thus, the least squares regression line is . The own price elasticity of demand for broilers is 1.38. From the t-statistic, this is statistically different from zero (the t-statistic is well over 2 in absolute value). The R-square is

relatively high, suggesting that the model explains 94 percent of the total variation in

the demand for chicken. Given that your current revenues are \$750,000 and the

elasticity of demand is 1.38, we may use the following formula to determine how

much you must change price to increase revenues by \$50,000:

Px，(;；RPQ1E xxQP,xxPx

Px，(;；\$50,000\$750,00011.38 Px

P\$50,000x0.175Solving yields . That is, to increase revenues by \$50,000, P\$285,000x

you must decrease your price by 17.5 percent.

Page 6 Michael R. Baye

17. The regression output (and corresponding demand equations) for each state are

presented below:

ILLINOIS

SUMMARY OUTPUT

Regression Statistics

Multiple R0.29

R Square0.09

Standard Error151.15

Observations50

ANOVA

degrees of freedomSSMSFSignificance F

Regression2100540.9350270.472.200.12

Residual471073835.1522847.56

Total491174376.08

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%

Intercept-42.65496.56-0.090.93-1041.60956.29

Price2.6213.990.190.85-25.5330.76

Income14.326.832.100.040.5828.05

Table 3-4

QPM，？？？42.652.6214.32The estimated demand equation is . While it appears that demand slopes upward, note that coefficient on price is not statistically different from zero. An increase in income by \$1,000 increases demand by 14.32 units. Since

the t-statistic associated with income is greater than 2 in absolute value, income is a

significant factor in determining quantity demanded. The R-square is extremely low, suggesting that the model explains only 9 percent of the total variation in the demand

for KBC microbrews. Factors other than price and income play an important role in determining quantity demanded.

Managerial Economics and Business Strategy, 4e Page 7

INDIANA

SUMMARY OUTPUT

Regression Statistics

Multiple R0.87

R Square0.76

Standard Error3.94

Observations50

ANOVA

degrees of freedomSSMSFSignificance F

Regression22294.931147.4673.960.00

Residual47729.1515.51

Total493024.08

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Intercept97.5310.888.960.0075.64119.42

Price-2.520.25-10.240.00-3.01-2.02

Income2.110.268.120.001.592.63

Table 3-5

Q97.532.52P2.11MThe estimated demand equation is . This equation says that increasing price by \$1 decreases quantity demanded by 2.52 units. Likewise,

increasing income by \$1,000 increases demand by 2.11 units. Since the t-statistics for each of the variables is greater than 2 in absolute value, price and income are

significant factors in determining quantity demanded. The R-square is reasonably high, suggesting that the model explains 76 percent of the total variation in the

demand for KBC microbrews.

Page 8 Michael R. Baye

MICHIGAN

SUMMARY OUTPUT

Regression Statistics

Multiple R0.63

R Square0.40

Standard Error10.59

Observations50

ANOVA

degrees of freedomSSMSFSignificance F

Regression23474.751737.3815.510.00

Residual475266.23112.05

Total498740.98

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Intercept182.4416.2511.230.0000149.75215.12

Price-1.020.31-3.280.0020-1.65-0.40

Income1.410.354.090.00020.722.11

Table 3-6

Q182.441.02P1.41MThe estimated demand equation is . This equation says that increasing price by \$1 decreases quantity demanded by 1.02 units. Likewise,

increasing income by \$1,000 increases demand by 1.41 units. Since the t-statistics associated with each of the variables is greater than 2 in absolute value, price and

income are significant factors in determining quantity demanded. The R-square is relatively low, suggesting that the model explains about 40 percent of the total

variation in the demand for KBC microbrews. The F-statistic is zero, suggesting that

the overall fit of the regression to the data is highly significant.

Managerial Economics and Business Strategy, 4e Page 9

MINNESOTA

SUMMARY OUTPUT

Regression Statistics

Multiple R0.64

R Square0.41

Standard Error16.43

Observations50

ANOVA

degrees of freedomSSMSFSignificance F

Regression28994.344497.1716.670.00

Residual4712680.48269.80

Total4921674.82

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%

Intercept81.7081.491.000.32-82.23245.62

Price-0.122.52-0.050.96-5.194.94 Income3.410.605.680.002.204.62

Table 3-7

QPM，？？81.700.123.41The estimated demand equation is . This equation says that increasing price by \$1 decreases quantity demanded by 0.12 units. Likewise, a

\$1,000 increase consumer income increases demand by 3.41 units. Since the t-statistic

associated with income is greater than 2 in absolute value, it is a significant factor in

determining quantity demanded. The R-square is relatively low, suggesting that the model explains 41 percent of the total variation in the demand for KBC microbrews.

Page 10 Michael R. Baye

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