managerial economics baye solutions (3-5)

36
Managerial Economics and Business Strategy, 7e Page 1 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. $0 $2 $4 $6 $8 $10 $12 $14 0 1 2 3 4 5 6 Quantity Price Demand MR Figure 3-1

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Solutions for Economics textbook. Baye, 8 edition.

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Page 1: Managerial Economics Baye Solutions (3-5)

Managerial Economics and Business Strategy, 7e Page 1

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.

$0

$2

$4

$6

$8

$10

$12

$14

0 1 2 3 4 5 6 Quantity

Price

Demand

MR

Figure 3-1

Page 2: Managerial Economics Baye Solutions (3-5)

Page 2 Michael R. Baye

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

1000 2 154 .02 400 700dxQ . Substituting the relevant information into

the elasticity formula gives: ,

1542 2 0.44

700x x

xQ P

x

PE

Q . Since this is less

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: 1000 2 354 .02 400 300d

xQ . Substituting the relevant information into

the elasticity formula gives: ,

3542 2 2.36

300x x

xQ P

x

PE

Q

. Since this is

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: 1000 2 154 .02 400 700d

xQ . Substituting the relevant information into

the elasticity formula gives: ,

400.02 .02 0.011

700x Z

ZQ P

x

PE

Q

. Since this

number is positive, goods X and Z are substitutes.

3. a. The own price elasticity of demand is simply the coefficient of ln Px, which is –

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 Py, which is –

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.

Page 3: Managerial Economics Baye Solutions (3-5)

Managerial Economics and Business Strategy, 7e Page 3

4.

a. Use the own price elasticity of demand formula to write %

25

dxQ . Solving,

we see that the quantity demanded of good X will decrease by 10 percent if the price of good X increases by 5 percent.

b. Use the cross-price elasticity of demand formula to write %

610

dxQ . Solving,

we see that the demand for X will decrease by 60 percent if the price of good Y increases by 10 percent.

c. Use the formula for the advertising elasticity of demand to write %

42

dxQ

.

Solving, we see that the demand for good X will decrease by 8 percent if advertising decreases by 2 percent.

d. Use the income elasticity of demand formula to write %

33

dxQ

. Solving, we

see that the demand of good X will decrease by 9 percent if income decreases by 3 percent.

5. Using the cross price elasticity formula, 5%

50

yP. Solving, we see that the price

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,

320$01.1.1000,70$5.21000,30$ R . Thus, a 1 percent increase in the price of good X would cause revenues from both goods to increase by $320.

Page 4: Managerial Economics Baye Solutions (3-5)

Page 4 Michael R. Baye

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

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.62R Square 0.39Adjusted R Square 0.37Standard Error 190.90Observations 100.00

ANOVA

degrees of freedom SS MS F Significance FRegression 2.00 2,223,017.77 1,111,508.88 30.50 0.00Residual 97.00 3,535,019.49 36,443.50Total 99.00 5,758,037.26

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 187.15 534.71 0.35 0.73 -880.56 1,254.86Price of X -4.32 0.69 6.26 0.00 -5.69 -2.96Income 0.09 0.02 4.47 0.00 0.05 0.14

Table 3-1

a. 187.15 4.32 .09d

x xQ P M .

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.

8. The approximate 95 percent confidence interval for a is 2102ˆ ˆ aa . Thus, you

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

95 percent confidence interval for b is 15.22ˆˆ b

b . Thus, you can be 95

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

Page 5: Managerial Economics Baye Solutions (3-5)

Managerial Economics and Business Strategy, 7e Page 5

9.

a. The t statistics are as follows: 848.007.11067

45.9369ˆ at ; 429.2

56.0

36.1ˆ b

t ; and

80.205.0

14.0ˆ

ct .

b. Since 2ˆ at the coefficient estimate, a , is not statistically different from zero.

However, since 2ˆ b

t and 2ˆ ct , the coefficient estimates b and c are

statistically different from zero. c. The R-square and adjust R-square tell us the proportion of variation explained by

the regression. The R-square tells us that 24 percent of the variability in the dependent variable is explained by price and income. The adjusted R-square confirms that fact and the R-square is not the result of estimating too many coefficients (i.e. few degrees of freedom).

10.

a. The own-price elasticity of demand is -1.36, so demand is elastic. b. The income elasticity of demandis-0.14, so X is an inferior good.

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

of demand for major cellular telephone manufacturer is 06.817

137,

PQE . Since

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

Page 6: Managerial Economics Baye Solutions (3-5)

Page 6 Michael R. Baye

12. The regression output is as follows:

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.97R Square 0.94Adjusted R Square 0.94Standard Error 0.00Observations 49

ANOVAdf SS MS F Significance F

Regression 2 0.00702 0.004 370.38 0.0000Residual 46 0.00044 0.000Total 48 0.00745

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 1.29 0.41 3.12 0.00 0.46 2.12LN Price -0.07 0.00 -26.62 0.00 -0.08 -0.07LN Income -0.03 0.09 -0.33 0.74 -0.22 0.16

Table 3-2

Thus, the demand for your batteries is given by ln 1.29 0.07 ln 0.03lnQ P M . 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.

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

5.12

3,

PQE . Thus, demand for Big G cereal is elastic (since this number is

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.

14. Use the income elasticity formula to write 75.14

%

dQ

. Solving, we see that coffee

purchases are expected to decrease by 7 percent.

Page 7: Managerial Economics Baye Solutions (3-5)

Managerial Economics and Business Strategy, 7e Page 7

15. To maximize revenue, Toyota should charge the price that makes demand unit elastic. Using the own price elasticity of demand formula,

, 1.25 1100,000 1.25Q P

PE

P

. Solving this equation for P implies that the

revenue maximizing price is 000,40$P . 16. Using the change in revenue formula for two products,

million 8.9$01.2.0400$5.21600$ R , so revenues will increase by $9.8 million.

17. The estimated demand function for residential heating fuel is

MPPPQ ENGRHFdRHF 05.092.1188.4369.9196.136 , where RHFP is the price

of residential heating fuel, NGP is the price of natural gas, EP is the price of

electricity, 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 RHFP is -91.69, a $2 increase in RHFP would lead to a 183.38 unit reduction in the consumption of residential heating fuel (since (-91.69)($2) = - 183.38 units). Since the coefficient of NGP is 43.88, a $1 reduction in NGP would lead to a

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.

Page 8: Managerial Economics Baye Solutions (3-5)

Page 8 Michael R. Baye

18. The regression output is as follows:

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.97R Square 0.94Adjusted R Square 0.94Standard Error 0.06Observations 41

ANOVAdf SS MS F Significance F

Regression 1 2.24 2.24 599.26 0.00Residual 39 0.15 0.00Total 40 2.38

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 4.29 0.12 37.17 0.00 4.06 4.53ln (Price) -1.38 0.06 -24.48 0.00 -1.50 -1.27

Table 3-3

Thus, the least squares regression line is ln 4.29 1.38 lnQ P . 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:

x

xPQxx P

PEQPR

xx

,1

x

x

P

P 38.11000,750$000,50$

Solving yields 175.0000,285$

000,50$

x

x

P

P. That is, to increase revenues by $50,000,

you must decrease your price by 17.5 percent.

Page 9: Managerial Economics Baye Solutions (3-5)

Managerial Economics and Business Strategy, 7e Page 9

19. The regression output (and corresponding demand equations) for each state are presented below:

ILLINOISSUMMARY OUTPUT

Regression StatisticsMultiple R 0.29R Square 0.09Adjusted R Square 0.05Standard Error 151.15Observations 50

ANOVA

degrees of freedom SS MS F Significance FRegression 2 100540.93 50270.47 2.20 0.12Residual 47 1073835.15 22847.56Total 49 1174376.08

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept -42.65 496.56 -0.09 0.93 -1041.60 956.29Price 2.62 13.99 0.19 0.85 -25.53 30.76Income 14.32 6.83 2.10 0.04 0.58 28.05

Table 3-4

The estimated demand equation is 42.65 2.62 14.32Q P M . 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.

Page 10: Managerial Economics Baye Solutions (3-5)

Page 10 Michael R. Baye

INDIANASUMMARY OUTPUT

Regression StatisticsMultiple R 0.87R Square 0.76Adjusted R Square 0.75Standard Error 3.94Observations 50

ANOVA

degrees of freedom SS MS F Significance FRegression 2 2294.93 1147.46 73.96 0.00Residual 47 729.15 15.51Total 49 3024.08

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 97.53 10.88 8.96 0.00 75.64 119.42Price -2.52 0.25 -10.24 0.00 -3.01 -2.02Income 2.11 0.26 8.12 0.00 1.59 2.63

Table 3-5

The estimated demand equation is MPQ 11.252.253.97 . 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 11: Managerial Economics Baye Solutions (3-5)

Managerial Economics and Business Strategy, 7e Page 11

MICHIGANSUMMARY OUTPUT

Regression StatisticsMultiple R 0.63R Square 0.40Adjusted R Square 0.37Standard Error 10.59Observations 50

ANOVA

degrees of freedom SS MS F Significance FRegression 2 3474.75 1737.38 15.51 0.00Residual 47 5266.23 112.05Total 49 8740.98

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 182.44 16.25 11.23 0.0000 149.75 215.12Price -1.02 0.31 -3.28 0.0020 -1.65 -0.40Income 1.41 0.35 4.09 0.0002 0.72 2.11

Table 3-6

The estimated demand equation is MPQ 41.102.144.182 . 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.

Page 12: Managerial Economics Baye Solutions (3-5)

Page 12 Michael R. Baye

MINNESOTASUMMARY OUTPUT

Regression StatisticsMultiple R 0.64R Square 0.41Adjusted R Square 0.39Standard Error 16.43Observations 50

ANOVA

degrees of freedom SS MS F Significance FRegression 2 8994.34 4497.17 16.67 0.00Residual 47 12680.48 269.80Total 49 21674.82

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 81.70 81.49 1.00 0.32 -82.23 245.62Price -0.12 2.52 -0.05 0.96 -5.19 4.94Income 3.41 0.60 5.68 0.00 2.20 4.62

Table 3-7

The estimated demand equation is 81.70 0.12 3.41Q P M . This equation says that increasing price by $1 decreases quantity demanded by 0.12 units. Likewise, a $1,000 increase in 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; however, price is not a statistically significant determinant of 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 13: Managerial Economics Baye Solutions (3-5)

Managerial Economics and Business Strategy, 7e Page 13

MISSOURISUMMARY OUTPUT

Regression StatisticsMultiple R 0.88R Square 0.78Adjusted R Square 0.77Standard Error 15.56Observations 50

ANOVA

degrees of freedom SS MS F Significance FRegression 2 39634.90 19817.45 81.81 0.00Residual 47 11385.02 242.23Total 49 51019.92

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 124.31 24.23 5.13 0.00 75.57 173.05Price -0.79 0.58 -1.36 0.18 -1.96 0.38Income 7.45 0.59 12.73 0.00 6.27 8.63

Table 3-8

The estimated demand equation is 124.31 0.79 7.45Q P M . This equation says that increasing price by $1 decreases quantity demanded by 0.79 units. Likewise, a $1,000 increase in income increases demand by 7.45 units. The t-statistic associated with price is not greater than 2 in absolute value; suggesting that price does not statistically impact the quantity demanded. However, the estimated income coefficient is statistically different from zero. The R-square is reasonably high, suggesting that the model explains 78 percent of the total variation in the demand for KBC microbrews.

Page 14: Managerial Economics Baye Solutions (3-5)

Page 14 Michael R. Baye

OHIOSUMMARY OUTPUT

Regression StatisticsMultiple R 0.99R Square 0.98Adjusted R Square 0.98Standard Error 10.63Observations 50

ANOVA

degrees of freedom SS MS F Significance FRegression 2 323988.26 161994.13 1434.86 0.00Residual 47 5306.24 112.90Total 49 329294.50

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 111.06 23.04 4.82 0.0000 64.71 157.41Price -2.48 0.79 -3.12 0.0031 -4.07 -0.88Income 7.03 0.13 52.96 0.0000 6.76 7.30

Table 3-9

The estimated demand equation is 111.06 2.48 7.03Q P M . This equation says that increasing price by $1 decreases quantity demanded by 2.48 units. Likewise, increasing income by $1,000 increases demand by 7.03 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 very high, suggesting that the model explains 98 percent of the total variation in the demand for KBC microbrews.

Page 15: Managerial Economics Baye Solutions (3-5)

Managerial Economics and Business Strategy, 7e Page 15

WISCONSINSUMMARY OUTPUT

Regression StatisticsMultiple R 0.999R Square 0.998Adjusted R Square 0.998Standard Error 4.79Observations 50

ANOVA

degrees of freedom SS MS F Significance FRegression 2 614277.37 307138.68 13369.30 0.00Residual 47 1079.75 22.97Total 49 615357.12

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 107.60 7.97 13.49 0.00 91.56 123.65Price -1.94 0.25 -7.59 0.00 -2.45 -1.42Income 10.01 0.06 163.48 0.00 9.88 10.13

Table 3-10

The estimated demand equation is 107.60 1.94 10.01Q P M . This equation says that increasing price by $1 decreases quantity demanded by 1.94 units. Likewise, increasing income by $1,000 increases demand by 10.01 units. Since the t-statistics associated with price and income are greater than 2 in absolute value, price and income are both significant factors in determining quantity demanded. The R-square is very high, suggesting that the model explains 99.8 percent of the total variation in the demand for KBC microbrews.

Page 16: Managerial Economics Baye Solutions (3-5)

Page 16 Michael R. Baye

20. Table 3-11 contains the output from the linear regression model. That model indicates that R2 = .55, or that 55 percent of the variability in the quantity demanded is explained by price and advertising. In contrast, in Table 3-12 the R2 for the log-linear model is .40, indicating that only 40 percent of the variability in the natural log of quantity is explained by variation in the natural log of price and the natural log of advertising. Therefore, the linear regression model appears to do a better job explaining variation in the dependent variable. This conclusion is further supported by comparing the adjusted R2s and the F-statistics in the two models. In the linear regression model the adjusted R2 is greater than in the log-linear model: .54 compared to .39, respectively. The F-statistic in the linear regression model is 58.61, which is larger than the F-statistic of 32.52 in the log-linear regression model. Taken together these three measures suggest that the linear regression model fits the data better than the log-linear model. Each of the three variables in the linear regression model is statistically significant; in absolute value the t-statistics are greater than two. In contrast, only two of the three variables are statistically significant in the log-linear model; the intercept is not statistically significant since the t-statistic is less than two in absolute value. At P = $3.10 and A = $100, milk consumption is 2.029 million gallons per week 029.2100005.10.361.152.6 d

milkQ .

SUMMARY OUTPUT LINEAR REGRESSION MODEL

Regression StatisticsMultiple R 0.74R Square 0.55Adjusted R Square 0.54Standard Error 1.06Observations 100.00

ANOVAdf SS MS F Significance F

Regression 2.00 132.51 66.26 58.61 2.05E-17Residual 97.00 109.66 1.13Total 99.00 242.17

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 6.52 0.82 7.92 0.00 4.89 8.15Price -1.61 0.15 -10.66 0.00 -1.92 -1.31Advertising 0.005 0.0016 2.96 0.00 0.00 0.01

Table 3-11

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Managerial Economics and Business Strategy, 7e Page 17

SUMMARY OUTPUT LOG-LINEAR REGRESSION MODEL

Regression StatisticsMultiple R 0.63R Square 0.40Adjusted R Square 0.39Standard Error 0.59Observations 100.00

ANOVAdf SS MS F Significance F

Regression 2.00 22.40 11.20 32.52 1.55E-11Residual 97.00 33.41 0.34Total 99.00 55.81

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept -1.99 2.24 -0.89 0.38 -6.44 2.46ln(Price) -2.17 0.28 -7.86 0.00 -2.72 -1.62ln(Advertising) 0.91 0.37 2.46 0.02 0.18 1.65

Table 3-12

21. Given the estimated demand function and the monthly subscriptions prices, demand is

172,000 subscribers 3010.13005.1509.05.152 dsatQ . Thus, revenues are

$8.6 million, which are not sufficient to cover costs. Revenues are maximized when

demand is unit elastic

19.217

9.sat

sat

P

P: Solving yields 56.120$satP . Thus, the

maximum revenue News Corp. can earn is $13,080,277.76 100056.1209.21756.120 QPTR . News Corp. cannot cover its costs

in the current environment. 22. The manager of Pacific Cellular estimated that the short-term price elasticity of

demand was inelastic. In the market for cellular service, contracts prevent many customers from immediately responding to price increases. Therefore, it is not surprising to observe inelastic in the short-term. However, as contracts expire and customers have more time to search for alternatives, quantity demanded is likely to drop off much more. Given a year or two, the demand for cellular service is much more elastic. The price increase has caused Pacific to lose more customers than they initially estimated.

23. The owner is confusing the demand for gasoline for the entire U.S. with demand for

the gasoline for individual gasoline stations. There are not a great number of substitutes for gasoline, but in large towns there are usually a very high number of substitutes for gasoline from an individual station. In order to make an informed decision, the owner needs to know the own price elasticity of demand for gasoline from his stations. Since gas prices are posted on big billboards, and gas stations in cities are generally close together, demand for gas from a small group of individual stations tends to be fairly elastic.

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Managerial Economics and Business Strategy, 7e Page 1

Chapter 4: Answers to Questions and Problems 1.

a. The market rate of substitution is 25.040

10

y

x

P

P.

b. See Figure 4-1. c. Increasing income to $800 (by $400) expands the budget set, as shown in Figure

4-1. Since the slope is unchanged, so is the market rate of substitution.

Budget Set

0

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 X

Y

Increase in income

Figure 4-1

2.

a. Since the slope of the line through point A is 20

120

and the price of good X

is $5, it follows that 5yP .

b. If the consumer spends all her income on good X she can purchase 20 units. Since these units cost $5 each, her income must be $100.

c. At point A, the consumer spends ($5)(10) = $50 on good Y, which means that the remaining $100 - $50 = $50 is being spent on good X. Since good X costs $5 per unit, point A corresponds to 10 units of good X.

d. The price of good Y decreased to $2.50. The consumer achieves a higher level of satisfaction at point B.

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Page 2 Michael R. Baye

3. a. The consumer’s budget line is YX 10$5$250$ . Rearranging terms and

solving for Y results in XY 5.025 . b. See in Figure 4-2. c. When the price of X increases to $10, the budget line becomes

YX 10$10$250$ , which is equivalent to XY 25 (after rearranging and simplifying terms). This is shown in Figure 4-2. The market rate of substitution

changes from 5 1

10 2x

y

P

P to

101

10x

y

P

P .

Budget Set

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35 40 45 50 X

Y

Figure 4-2

4. This is not always the case. For instance, if the consumer was initially consuming

more of the inferior good than a gift certificate would purchase, then less of the inferior good will be consumed when given a gift certificate.

5. A half-price sale cuts the price of each and every unit in half. In contrast, a buy-one,

get-one-free deal does not change the relative price of any units between 0 and 1 unit. Furthermore, it makes the price of units purchased between 1 and 2 units purchased zero.

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Managerial Economics and Business Strategy, 7e Page 3

6. a. 50$xP , 100$yP and M = $300.

b. 300

3100y

M

P units.

c. 300

650x

M

P units.

d. 1 unit (since the $50 gift certificate will purchase exactly one unit of good X).

e. $50 350

750x

M

P

units.

f. D , B, C, A. g. Normal.

7.

a. Consumption of good X will decrease and consumption of good Y will increase. b. Consumption of good X will decrease and consumption of good Y will increase. c. Nothing will happen to the consumption of either good. d. Consumption of good X will increase and consumption of good Y will decrease.

8. All properties hold except Property 4-3 (“Diminishing Marginal Rate of Substitution”) and Property 4-2 (“More is Better”).

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Page 4 Michael R. Baye

9. a. The initial budget set is depicted in Figure 4-3.

Figure 4-3

b. Doubling all income and price leaves the budget set unchanged. The increase in

income is sufficient to offset the price increases. The market rate of substitution is unchanged.

c. The consumer’s income is $500, the price of X is $2 per unit and the price of Y is $4 per unit.

10.

a. The workers opportunity set in a given 24-hour period is LE 5320 . b. Since the worker is always willing to trade $12 dollars of income for one hour of

leisure, the worker’s indifference curve does not exhibit diminishing marginal rate of substitution; the worker always trades between the two goods at the same rate.

11. These preferences do not exhibit a diminishing marginal rate of substitution since

consumers are always willing to substitute the same amount of store-brand sugar for an additional pound of producer-brand sugar. When store-brand sugar is $1 per pound and producer-brand sugar is $2 per pound, the consumer will purchase 10 pounds of store-label sugar and no producer-brand sugar. After the change, the consumer will purchase no store-label sugar and 10 pounds of producer-brand sugar.

Y

X

125

250

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Managerial Economics and Business Strategy, 7e Page 5

12. See Figure 4-6. When there is no food stamp program, the market rate of substitution is –0.5. The Food Stamp program leaves the market rate of substitution unchanged, and a consumer can purchase $170 of food without spending her income. A dollar-for-dollar exchange of food stamps for money further expands a consumer’s opportunity set, potentially making her better off.

Budget Constraint with and without Food Stamps

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70 80 90 100 110 120 130Food

Other Goods

Initial budget line

Budget line with $170 in food stamps

Budget line when food stamps are sold on black market for $170

Figure 4-6

13. See Figure 4-7. The offer expands the consumer’s budget set and allows her to

purchase more tires.

Budget Set with and without Buy 3, Get 4th Free Offer

0

100

200

300

400

500

600

0 1 2 3 4 5 6 7 8 9 10 11Tires

Income Spent on Other Goods

Budget line with "Buy 3, get the 4th Free Offfer"

Initial budget line

Figure 4-7

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Page 6 Michael R. Baye

14. See Figure 4-8. The initial market rate of substitution is –0.5. Since, after the price

decrease, the T

EM

P

PMRS 625.01 (where EMP is the price of electronic media

and TP the price of travel) equilibrium has not been achieved. To reach equilibrium, the business should increase its use of electronic media and decrease travel.

Budget Set

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10 Quantity of Electronic Media

Quantity of Travel

Initial budget line

New budget line

Figure 4-8

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15. The impacts on the consumer’s budget sets are illustrated in Figure 4-9. As is shown in the diagram, if the consumer has a strong preference for other goods (so that the preferred quantity of other goods is greater than 7 units), the cash is preferred even though it is taxed. Otherwise, the non-taxable, employer-sponsored health insurance program allows an employee to achieve a higher indifference curve.

Budget Line with Employer Sponsored Health Insurance

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8Quantity of

Health Insurance

Other Goods

Initial budget line

Budget line with healthinsurance benefit

Budget line with (taxable) cash equivalent health insurance benefit

Figure 4-9

16. Under the existing plan, a worker that does not “goof off” produces 3 copiers per hour

and thus is paid $9 each hour. Under the new plan, each worker would be paid a flat wage of $8 per hour. While it might appear on the surface that the company would save $1 per hour in labor costs by switching plans, the flat wage would be a lousy idea. Under the current plan, workers get paid the $9 only if they work hard during the hour and produce 3 machines that pass inspection. Under the new plan, workers would get paid $8 an hour regardless of how many units they produce. Since your firm has no supervisors to monitor the workers, you should not favor the plan.

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Page 8 Michael R. Baye

17. As shown in Figure 4-10, the budget line when more than 10 dozen bagels are purchased annually under the frequent buyer program is always greater than the budget line when the firm sells each dozen bagels at a 3 percent discount. However, the budget line for consumers who purchase fewer than 10 dozen bagels per year is greater under the 3 percent per dozen discount.

Comparison of Budget Lines Under Different Offers

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30 Quantity of Bagels (dozens)

Income Spent on Other Goods

Budget line with 3 percentdiscount

Budget line under the frequent buyer program

Figure 4-10

18. Yes. Since pizza is an inferior good, if the consumer is given $30 in cash she will

definitely spend it entirely on CDs – just as she would if given a $30 gift certificate at a local music store.

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Managerial Economics and Business Strategy, 7e Page 9

19. Figure 4-11 illustrates a consumer’s budget line when a firm offers a “quantity discount.” A consumer will never purchase exactly 8 bottles of wine, since at this kink in the opportunity set the consumer would always be better off by buying more or less wine.

Budget Line with Quantity Discount

0102030405060708090

100110

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Quantity of Wine

Quantity of Other Goods

Figure 4-11

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Page 10 Michael R. Baye

20. Figure 4-12 contains profit as a function of output. Output when managers are compensated based solely on output is 25 units and profits are zero. In contrast, when managers’ compensation is based solely on profits, output is 12.5 units and profits are $156.25. When managers’ compensation is based on a combination of output and profit, output ranges between 12.5 and 25 units and profit will be between zero and $156.25. The exact combination of output and profit depends on how these variables are weighted.

0

20

40

60

80

100

120

140

160

180

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5

Output (Q)

Profit ($)

Figure 4-12

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Managerial Economics and Business Strategy, 7e Page 11

21. Figures 4-13a and 4-13b, respectively, illustrate Albert’s and Sid’s opportunity sets.

Since there are 24 hours per day, at the new wage rate of $18 per hour Albert will supply 12 hours of labor per day (24-12), and Sid will supply 8 hours of labor per day (24-16). This seemingly contradictory result is explained by decomposing the wage change into the substitution effect and income effect. The diminishing marginal rate of substitution between income and leisure implies that the substitution effect will increase the amount of leisure consumed by each worker (decrease the amount of labor supplied). Since after the wage change Albert is observed consuming less leisure (supplying more labor), the income effect dominates the substitution effect. In contrast, the substitution effect dominates the income effect for Sid; since Sid is observed consuming more leisure (supplying less labor) after the wage change.

Figure 4-13a

Figure 4-13b

24Leisure

14

Income Albert’s Opportunity Set

12

432

480

24Leisure

14

Income Sid’s Opportunity Set

16

432

480

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Page 12 Michael R. Baye

22. Gift cards are not merely a fad. Retailers experience significant benefits from gift cards since they minimize product returns; independent of whether the good is normal or inferior. Gift cards can also benefit consumers. A gift card does not impact the amount purchased for one good (say the good on the Y axis), but shifts out the budget constraint for the other good (the good on the X axis) by the face value of the gift card. The expanded budget constrain permits the consumer to reach a higher indifference curve; resulting in greater utility.

23.

Under the Old Plan, consumers consumed 1,499 of online monthly minutes for $14.99. The budget line under the Flat-Rate Plan, however, is significantly different. Consumers can choose to now spend all their income on all other goods (AOG), represented by point A on the AOG axis or consume the same about AOG and any amount of online minutes up to the maximum number of minutes in a month. Optimizing consumers will choose the corner solution represented by the same number of units of AOG as the Old Plan and 43,200 online monthly minutes. Thus, UK consumers are necessarily better off (assuming no busy signals). AOL UK, however, gains no additional revenues and presumably must increase it network capacity. Therefore, AOL UK may earn lower profit (ignoring other factors).

AOG

43,200 1,499 43,200 1,499

AOG Old Plan

A

Flat-Rate Plan

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Managerial Economics and Business Strategy, 7e Page 1

Chapter 5: Answers to Questions and Problems 1.

a. When K = 16 and L = 16, 0.75 0.2516 16 16Q . Thus, APL = Q/L = 16/16 =

1. When K = 16 and L = 81, 0.75 0.2516 81 8 3 24Q . Thus, APL =

24/81 = 8/27.

b. The marginal product of labor is 3 42LMP L

. When L = 16,

3 42 16 1/ 4LMP

. When L = 81, 3 42 81 2 / 27LMP

. Thus, as the

number of units of labor hired increases, the marginal product of labor decreases 16 1/ 4 2 / 27 81L LMP MP , holding the level of capital fixed.

c. We must equate the value marginal product of labor equal to the wage and solve

for L. Here, 3/ 4 3/ 4$100 2 200L LVMP P MP L L

. Setting this

equal to the wage of $25 gives 3/ 4200 25L

. Solving for L, the optimal

quantity of labor is L = 16.

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Page 2 Michael R. Baye

2. See Table 5-1.

(1) (2) (3) (4) (5) (6) (7)

Capital Labor OutputMarginal

Product of Capital

Average Product of

Capital

Average Product of

Labor

Value Marginal Product of

CapitalK L Q MPK APK APL VMPK

0 20 0 -- -- -- --1 20 50 50 50 2.50 1002 20 150 100 75 7.50 2003 20 300 150 100 15 3004 20 400 100 100 20 2005 20 450 50 90 22.50 1006 20 475 25 79.17 23.75 507 20 475 0 67.86 23.75 08 20 450 -25 56.25 22.50 -509 20 400 -50 44.44 20 -10010 20 300 -100 30 15 -20011 20 150 -150 13.64 7.50 -300

Table 5-1

a. Labor is the fixed input while capital is the variable input. b. Fixed costs are 20($15) = $300. c. To produce 475 units in the least-cost manner requires 6 units of capital, which

cost $75 each. Thus, variable costs are ($75)(6) = $450. d. Using the VMPK = r rule, K = 5 maximizes profits. e. The maximum profits are $2(450) $15(20) $75(5) $225 . f. There are increasing marginal returns when K is between 0 and 3. g. There are decreasing marginal returns when K is between 3 and 11. h. There are negative marginal returns when K is greater than 7.

3. The law of diminishing marginal returns is the decline in marginal productivity

experienced when input usage increases, holding all other inputs constant. In contrast, the law of diminishing marginal rate of technical substitution is a property of a production function stating that as less of one input is used, increasing amounts of another input must be employed to produce the same level of output.

4.

a. FC = 50.

b. 2 310 25 10 30 10 5 10 $8, 250VC .

c. 300,8$105103010255010 32 C .

d. 5$10

50$10 AFC .

e. 10 $8, 25010 $825

10 10

VCAVC .

f. 830$101010 AVCAFCATC .

g. 125,2$101510602510 2 MC .

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5. Since r

wMRTS KL , the firm is not using the cost minimizing combination of labor

and capital. To minimize costs, the firm should use more labor and less capital since

the marginal product per dollar spent is greater for labor: 50 75

6 12L KMP MP

w r .

6. See Table 5-2.

(1) (2) (3) (4) (5) (6) (7) (8)

Quantity Fixed CostVariable

Cost Total CostAverage

Fixed Cost

Average Variable

CostAverage

Total CostMarginal

CostQ FC VC TC AFC AVC ATC MC0 10,000 0 10,000 -- -- -- --

100 10,000 10,000 20,000 100 100 200 100200 10,000 15,000 25,000 50 75 125 50300 10,000 30,000 40,000 33.33 100 133.33 150400 10,000 50,000 60,000 25 125 150 200500 10,000 90,000 100,000 20 180 200 400600 10,000 140,000 150,000 16.67 233.33 250 500

Table 5-2

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Page 4 Michael R. Baye

7. a. For a quadratic multi-product cost function, economies of scope exist if

021 QaQf . In this case, 75f and 25.0a , so economies of scope exist since f is fixed cost, which is always nonnegative.

b. Cost complementarities exist since 025.0 a . c. Since 025.0 a , the marginal cost of producing product 1 will increase if the

division that produces product 2 is sold. 8. Fixed costs are associated with fixed inputs, and do not change when output changes.

Variable costs are costs associated with variable inputs, and do change when output changes. Sunk costs are costs that are forever lost once they have been paid.

9.

a. When K = 2 and L = 3, Q = 4 units. b. The cost-minimizing mix of K and L that produce Q = 4 is K = 2, L = 1. c. Since K and L are perfect complements in the production process, the cost-

minimizing levels of K and L do not depend on the rental rates of K and L. Therefore, the cost-minimizing levels of K and L do not change with changes in the relative rental rates.

10.

a. With K = 2 and L = 3, Q = 16. b. Since the MRTSKL is 2, that means a company can trade two units of capital for

every one unit of labor. This production function does not exhibit diminishing marginal rate of technical substitution. The perfectly substitutability between capital and labor means that only input will be utilized. Since

10

2

30

4

r

MP

w

MP KL , the company should hire all capital.

c. The company should hire only labor.

11. An investment tax credit would reduce the relative price of capital to labor. Other

things equal, this would increase r

w, thereby making the isocost line more steep. This

means that the cost-minimizing input mix will now involve more capital and less labor, as firms substitute toward capital. Labor unions are likely to oppose the investment tax credit since the higher capital-to-labor ratio will translate into lost jobs. You might counter this argument by noting that, while some jobs will be lost due to substituting capital for labor, many workers will retain their jobs. Absent the plan, automakers have an incentive to substitute cheaper foreign labor for U.S. labor. The result of this substitution would be a movement of plants abroad, resulting in the complete loss of U.S. jobs.

12. Since r

wMRTSKL , the firm was not using the cost minimizing combination of labor

and capital. To achieve the cost minimizing combination of inputs, the previous

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Managerial Economics and Business Strategy, 7e Page 5

manager should have used fewer units of capital and more units of labor, since 100 100

8 16L KMP MP

w r .

13. The profit-maximizing level of labor and output is achieved where wVMPL . Here,

1/ 2 1 2 1/ 22 $100 4 $400LVMP L L

and 100$w per day. Solving yields L

= 16. The profit-maximizing level of output is 161642 2121 Q units. The firm’s fixed costs are $10,000, its variable costs are $100(16) = $1,600, and its total revenues are $200(16) = $3,200. Profits are $3,200 – $11,600 = – $8,400. The firm is suffering a loss, but the loss is lower than the $10,000 that would be lost if the firm shut down its operation.

14. The higher wage rate in Europe induces Airbus to employ a more capital intensive

input mix than Boeing. Since Airbus optimally uses fewer workers than Boeing, and profit-maximization entails input usage in the range of diminishing marginal product, it follows that the lower quantity of labor used by Airbus translates into a higher marginal product of labor at Airbus than at Boeing.

15. Table 5-3 provides some useful information for making your decision. According to the VMPL = w rule, you should hire five units of labor and produce 90 units of output to maximize profits. Your fixed costs are ($10)(5) = $50, your variable costs are ($50)(5) =$250, and your revenues are ($5)(90) = $450. Thus, your maximum profits are $450 - $300 = $150.

(1) (2) (3) (4) (5) (6) (7)

Labor Capital OutputMarginal

Product of Labor

Average Product of

Labor

Average Product of

Capital

Value Marginal Product of

LaborL K Q MPL APL APK VMPL

0 5 0 -- -- -- --1 5 10 10 10 2 502 5 30 20 15 6 1003 5 60 30 20 12 1504 5 80 20 20 16 1005 5 90 10 18 18 506 5 95 5 15.8 19 257 5 95 0 13.6 19 08 5 90 -5 11.3 18 -259 5 80 -10 8.9 16 -5010 5 60 -20 6 12 -10011 5 30 -30 2.7 6 -150

Table 5-3

16. The $1,200 per month that could be earned by renting out the excess rental space.

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Page 6 Michael R. Baye

17. Had she not spent the $6,000 on advertising but instead collected the $65,000 refund, her total loss would have been limited to her sunk costs of $10,000. Her decision to spend $6,000 on advertising in an attempt to fetch an extra $5,000 was clearly foolish. However, the $6,000 is a sunk cost and therefore irrelevant in deciding whether to accept the $66,000 offer. She should accept the $66,000 offer because doing so makes her $1,000 better off than obtaining the $65,000 refund.

18. Facility “L” produces 6 million kilowatt hours of electricity at the lowest average

total cost, so this is the optimal facility for South-Florida. Facility “M” produces 2 million kilowatt hours of electricity at the lowest average total cost, so this is the optimal facility for the Panhandle. There are economies of scale up to about 3 million kilowatts per hour, and diseconomies of scale thereafter. Therefore, facility “M” will be operating in the range of economies of scale while facility “L” will be operating in the range of diseconomies of scale.

19. To maximize profits the firm should continue adding workers so long as the value

marginal product of labor exceeds the wage. The value marginal product of labor is defined as the marginal product of labor times the price of output. Here, output sells for $50 per panel, so the value marginal product of the third worker is $50(290) = $14,500. Table 5-4 summarizes the VMPL for each choice of labor. Since the wage is $7,000, the profit maximizing number of workers is 4.

Machines Workers Output MPL VMPL Wage

5 0 0 – – – 5 1 600 600 $30,000 $7,000 5 2 1,000 400 $20,000 $7,000 5 3 1,290 290 $14,500 $7,000 5 4 1,480 190 $9,500 $7,000 5 5 1,600 120 $6,000 $7,000 5 6 1,680 80 $4,000 $7,000

Table 5-4

20. The rental rate of capital is ¥475,000, computed as

00,475000,9505. PMPr K . Therefore, the marginal product of labor is

0.0014 cars per hour, which is found by solving 000,475

5.0

330,1LMP

. Costs are

minimized when the marginal rate of technical substitution is 0.0028.

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Managerial Economics and Business Strategy, 7e Page 7

21. Given the tightly woven marine engine and shipbuilding divisions, economies of scope and cost complementarities are likely to exist. Eliminating the unprofitable marine engine division may actually raise the shipbuilding division’s costs and cause that division to become unprofitable. For this argument to withstand criticism, you must show the CEO that the quadratic multi-product cost function exhibits cost complementarities and economies of scope, which occurs when

0a and 021 QaQf , respectively, and compare profitability under the different scenarios.

22. Taking into account both implicit and explicit costs, the total fixed cost from

operating the kiosk is $6,000; the $2,000 in rent plus the $4,000 in forgone earnings. Total variable costs are $1.23 per gallon. The cost function is QQC 23.1000,6 .

The marginal cost is 23.1$

dQ

QdCQMC ; the wholesale price. The average

variable cost is 23.1$

23.1

Q

Q

Q

QCQAVC . The average fixed cost is

Q

QAFC6000$

. The entrepreneur will earn a profit when revenues exceed costs,

which occurs when QQ 23.1000,62 . Solving for Q implies the entrepreneur earns a profit when she sells Q > 8571.43 gallons, or 8572 gallons. The average fixed cost

of selling Q = 8572 is 70.0$8572

6000$8572 AFC .

23. Assuming that the optimal mix of unskilled and semi-skilled labor were being utilized

at the time the legislation passed, in the short run, a higher minimum wage paid to unskilled labor implies that to minimize costs the retailer should increase its use of semi-skilled worker and decrease its use or unskilled workers. In the longer run, the retailer may want to consider substituting capital for labor (invest in some machines to automate a portion of your boxing needs). Obviously, additional information would be required to conduct a net present value analysis for these long-run investments, but it is probably worth getting this information and running some numbers.