254065787-Managerial-Economics-Baye-Solutions-3-5

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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
$2
Demand
$0
0
1
2
3
MR 4
5
6 Quantity
Figure 3-1
Managerial Economics and Business Strategy, 7e
Page 1
2.
a. At the given prices, quantity demanded is 700 units:
Qxd  1000  2 154   .02  400   700 . Substituting the relevant information into
Px
154
 2
 0.44 . Since this is less
Qx
700
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:
Qxd  1000  2  354   .02  400   300 . Substituting the relevant information into
the elasticity formula gives: EQx , Px  2
P 
 354 
the elasticity formula gives: EQx , Px  2  x   2 
  2.36 . Since this is
 300 
 Qx 
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:
Qxd  1000  2 154   .02  400   700 . Substituting the relevant information into
P 
 400 
the elasticity formula gives: EQx , PZ  .02  Z   .02 
  0.011 . Since this
 700 
 Qx 
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 2
Michael R. Baye
4.
% Qxd
 2 . 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.
% Qxd
b. Use the cross-price elasticity of demand formula to write
 6 . Solving,
10
we see that the demand for X will decrease by 60 percent if the price of good Y
increases by 10 percent.
% Qxd
c. Use the formula for the advertising elasticity of demand to write
4.
2
Solving, we see that the demand for good X will decrease by 8 percent if
advertising decreases by 2 percent.
% Qxd
d. Use the income elasticity of demand formula to write
 3 . Solving, we
3
see that the demand of good X will decrease by 9 percent if income decreases by
3 percent.
a. Use the own price elasticity of demand formula to write
5.
6.
50
 5 . 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.
Using the cross price elasticity formula,
Using the change in revenue formula for two products,
R  $30,0001  2.5  $70,0001.1.01  $320 . Thus, a 1 percent increase in the
price of good X would cause revenues from both goods to increase by $320.
Managerial Economics and Business Strategy, 7e
Page 3
7.
Table 3-1 contains the answers to the regression output.
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.62
0.39
0.37
190.90
100.00
ANOVA
degrees of freedom
Regression
Residual
Total
2.00
97.00
99.00
Coefficients
Intercept
Price of X
Income
187.15
-4.32
0.09
SS
MS
F
2,223,017.77 1,111,508.88
3,535,019.49
36,443.50
5,758,037.26
Standard Error
t Stat
534.71
0.35
0.69
0.02
6.26
4.47
30.50
P-value
0.73
0.00
0.00
Significance F
0.00
Lower 95%
-880.56
-5.69
0.05
Upper 95%
1,254.86
-2.96
0.14
Table 3-1
a. Qxd  187.15  4.32 Px  .09 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.
Page 4
The approximate 95 percent confidence interval for a is aˆ  2 aˆ  10  2 . 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 bˆ  2 bˆ  2.5  1 . Thus, you can be 95
percent confident that b is within the range of –3.5 and –1.5.
Michael R. Baye
9.
a. The t statistics are as follows: t aˆ 
9369.45
1.36
 0.848 ; t bˆ 
 2.429 ; and
11067.07
0.56
 0.14
 2.80 .
0.05
b. Since t aˆ  2 the coefficient estimate, â , is not statistically different from zero.
t cˆ 
However, since t bˆ  2 and t cˆ  2 , the coefficient estimates b̂ and ĉ 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
137
of demand for major cellular telephone manufacturer is EQ ,P 
 8.06 . Since
 17
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.
Managerial Economics and Business Strategy, 7e
Page 5
12.
The regression output is as follows:
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.97
0.94
0.94
0.00
49
ANOVA
df
Regression
Residual
Total
Intercept
LN Price
LN Income
2
46
48
SS
0.00702
0.00044
0.00745
Coefficients Standard Error
1.29
0.41
-0.07
0.00
-0.03
0.09
MS
0.004
0.000
t Stat
3.12
-26.62
-0.33
F
Significance F
370.38
0.0000
P-value
0.00
0.00
0.74
Lower 95%
0.46
-0.08
-0.22
Upper 95%
2.12
-0.07
0.16
Table 3-2
Thus, the demand for your batteries is given by ln Q  1.29  0.07 ln P  0.03ln 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.
14.
Page 6
Based on this information, the own price elasticity of demand for Big G cereal is
3
EQ , P 
 1.5 . Thus, demand for Big G cereal is elastic (since this number is
2
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.
% Q d
 1.75 . Solving, we see that coffee
4
purchases are expected to decrease by 7 percent.
Use the income elasticity formula to write
Michael R. Baye
15.
To maximize revenue, Toyota should charge the price that makes demand unit elastic.
Using the own price elasticity of demand formula,
P


EQ , P   1.25 
  1 . Solving this equation for P implies that the
 100, 000  1.25P 
revenue maximizing price is P  $40,000 .
16.
Using the change in revenue formula for two products,
R  $6001  2.5  $400 0.2    .01  $9.8 million , so revenues will increase
by $9.8 million.
17.
The estimated demand function for residential heating fuel is
d
Q RHF
 136.96  91.69 PRHF  43.88PNG  11.92 PE  0.05M , where PRHF is the price
of residential heating fuel, PNG is the price of natural gas, PE 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 PRHF is -91.69, a $2 increase in PRHF 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 PNG is 43.88, a $1 reduction in PNG 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.
Managerial Economics and Business Strategy, 7e
Page 7
18.
The regression output is as follows:
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.97
0.94
0.94
0.06
41
ANOVA
df
Regression
Residual
Total
Intercept
ln (Price)
SS
1
39
40
2.24
0.15
2.38
MS
2.24
0.00
F
Significance F
599.26
0.00
Coefficients Standard Error t Stat P-value
4.29
0.12 37.17
0.00
-1.38
0.06 -24.48
0.00
Lower 95%
Upper 95%
4.06
4.53
-1.50
-1.27
Table 3-3
Thus, the least squares regression line is ln Q  4.29  1.38 ln 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:


Px
Px
P
$50,000  $750,0001  1.38 x
Px

R  Px  Q x 1  EQx ,Px 
Px
$50,000

 0.175 . That is, to increase revenues by $50,000,
Px
 $285,000
you must decrease your price by 17.5 percent.
Solving yields
Page 8
Michael R. Baye
19.
The regression output (and corresponding demand equations) for each state are
presented below:
ILLINOIS
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.29
0.09
0.05
151.15
50
ANOVA
degrees of freedom
Regression
Residual
Total
2
47
49
SS
MS
F
Significance F
100540.93
1073835.15
1174376.08
50270.47
22847.56
2.20
0.12
t Stat
P-value
Lower 95%
Coefficients Standard Error
Intercept
Price
Income
-42.65
2.62
14.32
496.56
13.99
6.83
-0.09
0.19
2.10
0.93
0.85
0.04
-1041.60
-25.53
0.58
Upper 95%
956.29
30.76
28.05
Table 3-4
The estimated demand equation is Q  42.65  2.62 P  14.32 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.
Managerial Economics and Business Strategy, 7e
Page 9
INDIANA
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.87
0.76
0.75
3.94
50
ANOVA
degrees of freedom
Regression
Residual
Total
2
47
49
SS
MS
2294.93 1147.46
729.15
15.51
3024.08
Coefficients Standard Error
Intercept
Price
Income
97.53
-2.52
2.11
10.88
0.25
0.26
t Stat
8.96
-10.24
8.12
F
73.96
P-value
0.00
0.00
0.00
Significance F
0.00
Lower 95%
75.64
-3.01
1.59
Upper 95%
119.42
-2.02
2.63
Table 3-5
The estimated demand equation is Q  97.53  2.52 P  2.11M . 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 10
Michael R. Baye
MICHIGAN
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.63
0.40
0.37
10.59
50
ANOVA
degrees of freedom
Regression
Residual
Total
2
47
49
SS
MS
F
Significance F
3474.75
5266.23
8740.98
1737.38
112.05
15.51
0.00
t Stat
P-value
Lower 95%
Upper 95%
11.23
-3.28
4.09
0.0000
0.0020
0.0002
149.75
-1.65
0.72
215.12
-0.40
2.11
Coefficients Standard Error
Intercept
Price
Income
182.44
-1.02
1.41
16.25
0.31
0.35
Table 3-6
The estimated demand equation is Q  182.44  1.02 P  1.41M . 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, 7e
Page 11
MINNESOTA
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.64
0.41
0.39
16.43
50
ANOVA
degrees of freedom
Regression
Residual
Total
2
47
49
SS
MS
F
Significance F
8994.34
12680.48
21674.82
4497.17
269.80
16.67
0.00
t Stat
P-value
Lower 95%
Upper 95%
1.00
-0.05
5.68
0.32
0.96
0.00
-82.23
-5.19
2.20
245.62
4.94
4.62
Coefficients Standard Error
Intercept
Price
Income
81.70
-0.12
3.41
81.49
2.52
0.60
Table 3-7
The estimated demand equation is Q  81.70  0.12 P  3.41M . 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 tstatistic 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 12
Michael R. Baye
MISSOURI
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.88
0.78
0.77
15.56
50
ANOVA
degrees of freedom
Regression
Residual
Total
2
47
49
SS
MS
F
Significance F
39634.90
11385.02
51019.92
19817.45
242.23
81.81
0.00
t Stat
P-value
Lower 95%
Upper 95%
5.13
-1.36
12.73
0.00
0.18
0.00
75.57
-1.96
6.27
173.05
0.38
8.63
Coefficients Standard Error
Intercept
Price
Income
124.31
-0.79
7.45
24.23
0.58
0.59
Table 3-8
The estimated demand equation is Q  124.31  0.79 P  7.45M . 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.
Managerial Economics and Business Strategy, 7e
Page 13
OHIO
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.99
0.98
0.98
10.63
50
ANOVA
degrees of freedom
Regression
Residual
Total
2
47
49
SS
323988.26
5306.24
329294.50
Coefficients Standard Error
Intercept
Price
Income
111.06
-2.48
7.03
23.04
0.79
0.13
MS
F
161994.13 1434.86
112.90
Significance F
0.00
t Stat
P-value
Lower 95%
Upper 95%
4.82
-3.12
52.96
0.0000
0.0031
0.0000
64.71
-4.07
6.76
157.41
-0.88
7.30
Table 3-9
The estimated demand equation is Q  111.06  2.48P  7.03M . 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 14
Michael R. Baye
WISCONSIN
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.999
0.998
0.998
4.79
50
ANOVA
degrees of freedom
Regression
Residual
Total
2
47
49
SS
Coefficients Standard Error
Intercept
Price
Income
107.60
-1.94
10.01
MS
F
614277.37 307138.68 13369.30
1079.75
22.97
615357.12
7.97
0.25
0.06
t Stat
13.49
-7.59
163.48
Significance F
0.00
P-value
Lower 95%
Upper 95%
0.00
0.00
0.00
91.56
-2.45
9.88
123.65
-1.42
10.13
Table 3-10
The estimated demand equation is Q  107.60  1.94 P  10.01M . 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.
Managerial Economics and Business Strategy, 7e
Page 15
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
d
gallons per week Qmilk
 6.52  1.613.10  .005100   2.029 .


SUMMARY OUTPUT LINEAR REGRESSION MODEL
Regression Statistics
Multiple R
0.74
R Square
0.55
Adjusted R Square
0.54
Standard Error
1.06
Observations
100.00
ANOVA
df
Regression
Residual
Total
Intercept
Price
Advertising
2.00
97.00
99.00
SS
132.51
109.66
242.17
MS
66.26
1.13
F
Significance F
58.61
2.05E-17
Coefficients Standard Error t Stat P-value
6.52
0.82
7.92
0.00
-1.61
0.15 -10.66
0.00
0.005
0.0016
2.96
0.00
Lower 95%
Upper 95%
4.89
8.15
-1.92
-1.31
0.00
0.01
Table 3-11
Page 16
Michael R. Baye
SUMMARY OUTPUT LOG-LINEAR REGRESSION MODEL
Regression Statistics
Multiple R
0.63
R Square
0.40
Adjusted R Square
0.39
Standard Error
0.59
Observations
100.00
ANOVA
df
Regression
Residual
Total
SS
2.00
97.00
99.00
MS
22.40 11.20
33.41 0.34
55.81
F
Significance F
32.52
1.55E-11
Coefficients Standard Error t Stat P-value
-1.99
2.24 -0.89
0.38
-2.17
0.28 -7.86
0.00
0.91
0.37 2.46
0.02
Intercept
ln(Price)
ln(Advertising)
Lower 95%
Upper 95%
-6.44
2.46
-2.72
-1.62
0.18
1.65
Table 3-12
21.
Given the estimated demand function and the monthly subscriptions prices, demand is
d
172,000 subscribers Qsat
 152.5  0.950  1.0530  1.1030 . Thus, revenues are
$8.6 million, which are not sufficient to cover costs. Revenues are maximized when
 
 
Psat
  1 : Solving yields Psat  $120.56 . Thus, the
demand is unit elastic  .9

217

.
9
P
sat
 
 
maximum revenue News Corp. can earn is $13,080,277.76
TR  P  Q  120.56  217  .9  120.56  1000 . 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.

Managerial Economics and Business Strategy, 7e

Page 17
Chapter 4: Answers to Questions and Problems
1.
a. The market rate of substitution is 
Px
10

 0.25 .
40
Py
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
Y
25
20
15
Increase
in income
10
5
0
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
X
Figure 4-1
2.
a. Since the slope of the line through point A is 
20
 1 and the price of good X
20
is $5, it follows that Py  5 .
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.
Managerial Economics and Business Strategy, 7e
Page 1
3.
a. The consumer’s budget line is $250  $5 X  $10Y . Rearranging terms and
solving for Y results in Y  25  0.5 X .
b. See in Figure 4-2.
c. When the price of X increases to $10, the budget line becomes
$250  $10 X  $10Y , which is equivalent to Y  25  X (after rearranging and
simplifying terms). This is shown in Figure 4-2. The market rate of substitution
P
P
5
1
10
changes from  x     to  x    1 .
10
2
Py
Py
10
Budget Set
Y
30
25
20
15
10
5
0
0
5
10
15
20
25
30
35
40
45
50
X
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.
Page 2
Michael R. Baye
6.
a. Px  $50 , Py  $100 and M = $300.
b.
M 300

 3 units.
Py 100
f.
g.
M 300

 6 units.
Px
50
1 unit (since the $50 gift certificate will purchase exactly one unit of good X).
M  $50 350

 7 units.
Px
50
D , B, C, A.
Normal.
a.
b.
c.
d.
Consumption of good X will decrease and consumption of good Y will increase.
Consumption of good X will decrease and consumption of good Y will increase.
Nothing will happen to the consumption of either good.
Consumption of good X will increase and consumption of good Y will decrease.
c.
d.
e.
7.
8.
All properties hold except Property 4-3 (“Diminishing Marginal Rate of
Substitution”) and Property 4-2 (“More is Better”).
Managerial Economics and Business Strategy, 7e
Page 3
9.
a. The initial budget set is depicted in Figure 4-3.
Y
125
Figure 4-3
250
X
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.
11.
Page 4
a. The workers opportunity set in a given 24-hour period is E  320  5L .
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.
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.
Michael R. Baye
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 dollarfor-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
Other
80
Goods
Budget line when food stamps are sold on black market for $170
70
60
50
Budget line with $170 in food stamps
40
30
20
Initial budget line
10
0
0
10 20 30 40 50 60
70 80 90 100 110 120 130
Food
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
Income Spent on
Other
600
Goods
Budget line with "Buy 3, get the 4th Free Offfer"
500
400
300
200
Initial budget line
100
0
0
1
2
3
4
5
6
7
8
9
10
11
Tires
Figure 4-7
Managerial Economics and Business Strategy, 7e
Page 5
14.
See Figure 4-8. The initial market rate of substitution is –0.5. Since, after the price
P
decrease, the MRS  1  0.625   EM (where PEM is the price of electronic media
PT
and PT 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
Quantity
of Travel
7
6
5
New budget line
4
3
2
Initial budget line
1
0
0
1
2
3
4
5
6
7
8
9
10 Quantity of
Electronic Media
Figure 4-8
Page 6
Michael R. Baye
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
Other Goods
9
Budget line with (taxable) cash
equivalent health insurance benefit
8
7
6
5
Budget line with health
insurance benefit
4
3
2
Initial budget line
1
0
0
1
2
3
4
5
6
7
8
Quantity of
Health Insurance
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.
Managerial Economics and Business Strategy, 7e
Page 7
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
Income Spent
on Other Goods 160
140
120
100
Budget line under the
frequent buyer program
80
60
Budget line with 3 percent
discount
40
20
0
0
5
10
15
20
25
30
Quantity of
Bagels (dozens)
Figure 4-10
18.
Page 8
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.
Michael R. Baye
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
Quantity of
Other Goods
110
100
90
80
70
60
50
40
30
20
10
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Quantity
of Wine
Figure 4-11
Managerial Economics and Business Strategy, 7e
Page 9
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
$156.25. The exact combination of output and profit depends on how these variables
are weighted.
Profit ($) 180
160
140
120
100
80
60
40
20
0
0
2.5
5
7.5
10
12.5
15
17.5
20
22.5
25
27.5
Output (Q)
Figure 4-12
Page 10
Michael R. Baye
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.
Income
Albert’s Opportunity Set
480
432
12
14
24
Leisure
Figure 4-13a
Income
Sid’s Opportunity Set
480
432
14
16
24
Leisure
Figure 4-13b
Managerial Economics and Business Strategy, 7e
Page 11
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.
AOG
Flat-Rate Plan
Old Plan
AOG
A
1,499
43,200
1,499
43,200
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).
Page 12
Michael R. Baye
Chapter 5: Answers to Questions and Problems
1.
a. When K = 16 and L = 16, Q  16 
16   16 . Thus, APL = Q/L = 16/16 =
0.75
0.25
1. When K = 16 and L = 81, Q  16   81  8   3  24 . Thus, APL =
0.75
0.25
24/81 = 8/27.
3 4
b. The marginal product of labor is MPL  2  L  . When L = 16,
MPL  2 16 
3 4
 1/ 4 . When L = 81, MPL  2 81
3 4
 2 / 27 . Thus, as the
number of units of labor hired increases, the marginal product of labor decreases
MPL 16   1/ 4  2 / 27  MPL  81 , 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, VMPL   P  MPL    $100  2  L 
equal to the wage of $25 gives 200  L 
quantity of labor is L = 16.
3/ 4
Managerial Economics and Business Strategy, 7e
3/ 4
  200  L 
3/ 4
. Setting this
 25 . Solving for L, the optimal
Page 1
2. See Table 5-1.
(1)
(2)
(3)
Q
(4)
Marginal
Product of
Capital
MP K
(5)
Average
Product of
Capital
AP K
(6)
Average
Product of
Labor
AP L
(7)
Value Marginal
Product of
Capital
VMPK
Capital
Labor
Output
K
L
0
1
2
3
4
5
6
7
8
9
10
11
20
20
20
20
20
20
20
20
20
20
20
20
0
50
150
300
400
450
475
475
450
400
300
150
-50
100
150
100
50
25
0
-25
-50
-100
-150
-50
75
100
100
90
79.17
67.86
56.25
44.44
30
13.64
-2.50
7.50
15
20
22.50
23.75
23.75
22.50
20
15
7.50
-100
200
300
200
100
50
0
-50
-100
-200
-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.
2
3
b. VC 10   25 10   30 10   5 10   $8, 250 .
c. C 10  50  2510  3010  510  $8,300 .
$50
d. AFC 10  
 $5 .
10
VC 10  $8, 250

 $825 .
e. AVC 10  
10
10
f. ATC 10   AFC 10   AVC 10   $830 .
2
3
g. MC 10   25  6010   1510   $2,125 .
2
Page 2
Michael R. Baye
w
, the firm is not using the cost minimizing combination of labor
r
and capital. To minimize costs, the firm should use more labor and less capital since
MPL 50 MPK 75
the marginal product per dollar spent is greater for labor:
.



6
12
w
r
5.
Since MRTS KL 
6.
See Table 5-2.
(1)
(2)
(3)
(4)
(5)
(7)
(8)
Average
Fixed Cost
(6)
Average
Variable
Cost
Quantity
Fixed Cost
Variable
Cost
Total Cost
Average
Total Cost
Marginal
Cost
Q
FC
VC
0
100
200
300
400
500
600
10,000
10,000
10,000
10,000
10,000
10,000
10,000
0
10,000
15,000
30,000
50,000
90,000
140,000
TC
AFC
AVC
ATC
MC
10,000
20,000
25,000
40,000
60,000
100,000
150,000
-100
50
33.33
25
20
16.67
-100
75
100
125
180
233.33
-200
125
133.33
150
200
250
-100
50
150
200
400
500
Table 5-2
Managerial Economics and Business Strategy, 7e
Page 3
7.
a. For a quadratic multi-product cost function, economies of scope exist if
f  aQ1Q2  0 . In this case, f  75 and a  0.25 , so economies of scope exist
since f is fixed cost, which is always nonnegative.
b. Cost complementarities exist since a  0.25  0 .
c. Since a  0.25  0 , 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 costminimizing 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
MPL MPK
4
2


 , the company should hire all capital.
w
r
30 10
c. The company should hire only labor.
11.
An investment tax credit would reduce the relative price of capital to labor. Other
w
things equal, this would increase , thereby making the isocost line more steep. This
r
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 MRTS KL 
Page 4
w
, the firm was not using the cost minimizing combination of labor
r
and capital. To achieve the cost minimizing combination of inputs, the previous
Michael R. Baye
manager should have used fewer units of capital and more units of labor, since
MPL 100 MPK 100



.
w
r
8
16
13.
The profit-maximizing level of labor and output is achieved where VMPL  w . Here,
VMPL  2  $100   4 
1/ 2
 L
1 2
 $400  L 
1/ 2
and w  $100 per day. Solving yields L
= 16. The profit-maximizing level of output is Q  24  16  16 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.
12
12
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)
Q
(4)
Marginal
Product of
Labor
MP L
(5)
Average
Product of
Labor
AP L
(6)
Average
Product of
Capital
AP K
(7)
Value Marginal
Product of
Labor
VMPL
Labor
Capital
Output
L
K
0
1
2
3
4
5
6
7
8
9
10
11
5
5
5
5
5
5
5
5
5
5
5
5
0
10
30
60
80
90
95
95
90
80
60
30
-10
20
30
20
10
5
0
-5
-10
-20
-30
-10
15
20
20
18
15.8
13.6
11.3
8.9
6
2.7
-2
6
12
16
18
19
19
18
16
12
6
-50
100
150
100
50
25
0
-25
-50
-100
-150
Table 5-3
16.
The $1,200 per month that could be earned by renting out the excess rental space.
Managerial Economics and Business Strategy, 7e
Page 5
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
5
5
5
5
5
5
5
Workers Output MPL VMPL Wage
0
0
–
–
–
1
600
600 $30,000 $7,000
2
1,000
400 $20,000 $7,000
3
1,290
290 $14,500 $7,000
4
1,480
190 $9,500 $7,000
5
1,600
120 $6,000 $7,000
6
1,680
80
$4,000 $7,000
Table 5-4
20.
Page 6
The rental rate of capital is ¥475,000, computed as
r  MPK  P  .5  950,000  475,00 . Therefore, the marginal product of labor is
MPL
0.5
0.0014 cars per hour, which is found by solving
. Costs are

1,330 475,000
minimized when the marginal rate of technical substitution is 0.0028.
Michael R. Baye
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
a  0 and f  aQ1Q2  0 , 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 C Q   6,000  1.23Q .
dC Q 
 $1.23 ; the wholesale price. The average
The marginal cost is MC Q  
dQ
C Q  1.23Q

 $1.23 . The average fixed cost is
variable cost is AVC Q  
Q
Q
$6000
AFC Q  
. The entrepreneur will earn a profit when revenues exceed costs,
Q
which occurs when 2Q  6,000  1.23Q . Solving for Q implies the entrepreneur earns
a profit when she sells Q > 8571.43 gallons, or 8572 gallons. The average fixed cost
$6000
of selling Q = 8572 is AFC 8572 
 $0.70 .
8572
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.
Managerial Economics and Business Strategy, 7e
Page 7
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