Student Practice Manual
to Accompany
Microeconomics
B. Douglas Bernheim
Michael Whinston
Created by:
Jennifer Pate, Ph.D.
Loyola Marymount University
Table of Contents
Chapter 1
Page
3
Additional Exercises for Chapter 1
4
Chapter 2
5
Additional Exercises for Chapter 2
7
Chapter 3
11
Additional Exercises for Chapter 3
14
Chapter 4
18
Additional Exercises for Chapter 4
21
Chapter 5
27
Additional Exercises for Chapter 5
33
Chapter 6
44
Additional Exercises for Chapter 6
51
Chapter 7
57
Additional Exercises for Chapter 7
59
Chapter 8
65
Additional Exercises for Chapter 8
69
Chapter 9
73
Additional Exercises for Chapter 9
78
Chapter 10
82
Additional Exercises for Chapter 10
85
Chapter 11
89
Additional Exercises for Chapter 11
93
Chapter 12
99
Additional Exercises for Chapter 12
02
Chapter 13
09
Additional Exercises for Chapter 13
11
Chapter 14
15
Additional Exercises for Chapter 14
17
Chapter 15
22
Additional Exercises for Chapter 15
24
Chapter 16
28
Additional Exercises for Chapter 16
30
Chapter 17
34
Additional Exercises for Chapter 17
36
Chapter 18
41
Additional Exercises for Chapter 18
44
Chapter 19
50
Additional Exercises for Chapter 19
52
Chapter 20
55
Additional Exercises for Chapter 20
57
Chapter 1
Sample Problems
. Government intervention in markets is a politically divisive issue. What are some potential justifications of government intervention in markets?
Answer:
Government intervention in markets is potentially justifiable when market failures occur. In a case where there is little competition in a market, the government could step in and regulate prices so that consumers are not unjustly hurt by lack of competition in a market. The government could also intervene in markets to bridge the gap between buyers and sellers created by a lack of information. In such cases, the government could impose regulations and mandates on disclosure of information. Also, inequality across customers in a market could justify a government to implement policies to redistribute resources more equitably.
2. Simplifying assumptions are important when creating an economic model because it helps economists focus on the most important reasons for a phenomenon. How does assuming people are motivated by material self-interest simplify an economic model?
Answer:
Assuming material self-interest simplifies economic models because it means that everyone is motivated by the same principles. There are no altruistic individuals. In short it makes everyone similar and that allows models to be more broadly applicable.
3. Your roommate asks you for advice about what to do this Friday night. She has narrowed her options to two different choices. She could go to a party with her new friends, or she could attend an off-campus seminar on The Poetry of Robert Frost which her English professor recommended as an event she would truly enjoy. To ease her decision-making process, explain the trade-offs between her two choices.
Answer:
If she chooses to go to the party she will forgo the opportunity to experience the poetry of Robert Frost which was highly recommended to her. If she goes to the seminar instead of the party she will forgo a chance to bond with her new friends and the possibility of having a good time. Her decision will involve an opportunity cost equal to the forgone action.
Additional Exercises for Chapter 1
. Which type of economy, communist or capitalist, is more closely described as a free market system and why?
2. What are markets and why do they require property rights?
3. How does the principle of individual sovereignty allow economists to conduct normative analysis?
4. How does the field of economics go beyond the study of money?
5. Why is it important to identify additional implications after formulating a theory?
6. What is the benefit to economists of using simplifying assumptions when formulating a model?
7. List and briefly describe the three main categories of economic data.
8. Why might economists disagree about positive matters?
9. How does a natural experiment differ from a controlled experiment?
0. Given that people respond to incentives, how do changes in prices alter behavior?
1. How can trade benefit two people at the same time?
2. Why might it be useful to use economics to evaluate environmental policy?
Chapter 2
Sample Problems
. Assume the demand for ice cream takes the following form:
If the price of yogurt is $4.00, the price of a waffle cone is $3.00 and the average income is $32,000, what is the demand function for ice cream? Graph it below.
Answer:
To solve for the demand function, substitute in the prices for yogurt and waffle cones and the average income, then simplify as in the equations below:
To graph the equation, first find the endpoints. If, then and when, .
2. Using the information from the previous question, at what will the quantity demanded of ice cream equal 18 gallons? How will this change if the price of waffle cones decreases to $2.00?
Answer:
To find the price of ice cream where quantity demanded equals 18 gallons, set the equation for quantity demanded to 18 and solve for.
If the price of waffle cones falls to $2.00, the demand function changes to
To again find the price of ice cream where quantity demanded equals 18 gallons, set the equation for quantity demanded to 18 and solve for.
3. If this means that if, then and if, then. What is the elasticity of demand at a price of $3.00? What about at a price of $4.00?
Answer:
At $3.00:
At $4.00:
Additional Exercises for Chapter 2
. Assume the demand for French fries takes the following form:
If the price of onion rings is $5.00, the price of ketchup is $4.00, and the average income is $30,000, what is the demand function for French fries? Graph it below.
2. Following from the information in question (1), at whatwill the quantity demanded equal 15 million pounds? How will this change if the price of ketchup increases to $5.00?
3. What happens to the demand curve for French fries if the average income increases to $40,000, holding all else constant? Graph this new demand curve alongside your answer from (1).
4. Given the demand function in (1), how can you determine whether French fries and onion rings are substitutes or complements? Explain.
5.
Assume the supply of French fries (FF) takes the following form:
If the price of frying oil is $3.00 and the price of potato flakes is $5.00, what is the supply function for French fries? Graph it below.
6. Following from (5), at whatwill the quantity supplied equal 12.5 million pounds? How will this change if the price of potato flakes falls to $3.00?
7. How does your answer from (6) demonstrate that French fries and potato flakes are substitutes in production? Explain.
8. What happens to the supply curve for French fries if the price of frying oil increases to $4.50? Graph this new supply curve alongside your answer from (5).
9. Assume the demand function for French fries takes the form: and the supply function takes the form: . What is the equilibrium price in this market? What are the amounts bought and sold?
0. Following from (9), what happens to the equilibrium price and quantity if demand decreases to? Calculate the new equilibrium.
1. Starting again from (9), what happens to the equilibrium price and quantity if supply increases to? Calculate the new equilibrium.
2. Finally, what happens to the equilibrium price and quantity if demand decreases to and supply increases to? Calculate the new equilibrium. Does quantity increase, decrease, or stay the same and why?
3. Suppose the demand function for ice cream is: and the supply function is: Find the equilibrium price in the market for ice cream. What are the amounts that are bought and sold? Graph the supply and demand functions below.
4. If, this means that if, then and if, then. What is the elasticity of demand at a price of $3.50? What about at a price of $5.00?
5. If, this means that if, then and if, then. What is the elasticity of supply at a price of $3.50? What about at a price of $5.00?
6. Considering the demand function, at what price will total expenditure be the largest?
Chapter 3
Sample Problems
. In an effort to help him avoid the freshman 15, Jerry's parents bought him a gym membership. To keep himself motivated Jerry wants to arrange personal trainer sessions for the semester. Jerry can hire the trainer for up to 21 sessions in increments of 3 sessions. The total benefit (TB) and total cost (TC) of the training sessions are in the table below. Calculate the net benefit (NB) for each of Jerry's choices and use your answer to identify his best option.
Number of Sessions
Benefit
Cost
Net Benefit
0
0
0
3
15
90
6
245
74
9
396
252
2
525
324
5
563
390
8
608
450
21
649
504
Answer:
Number
of Sessions
Benefit
Cost
Net Benefit
0
0
0
0
3
15
90
25
6
245
74
71
9
396
252
44
2
525
324
201
5
563
390
73
8
608
450
58
21
649
504
45
The best choice for Jerry would be to hire the personal trainer for the number of sessions that yield the highest net benefit. According to the table Jerry's best choice is to hire the personal trainer for 12 sessions with a net benefit of 201.
2. Calculate the marginal benefit (MB) and marginal cost (MC) to fill in the table below. How can you use your answers to identify the best choice?
Number
of Sessions
Benefit
Marginal Benefit
Cost
Marginal Cost
0
0
0
3
15
90
6
245
74
9
396
252
2
525
324
5
563
390
8
608
450
21
649
504
Answer:
Number
of Sessions
Benefit
Marginal Benefit
Cost
Marginal Cost
0
0
X
0
X
3
15
15
90
90
6
245
30
74
84
9
396
51
252
78
2
525
29
324
72
5
563
38
390
66
8
608
45
450
60
21
649
41
504
54
The best choice for Jerry according to this schedule is to hire a personal trainer for twelve sessions. The twelve session mark is the last point at which the marginal benefit exceeds the marginal cost.
3. Suppose that Jerry can hire a personal trainer for up to twenty-one sessions. The total benefit of hiring the trainer is given by the functionand the total cost is given by The corresponding functions for MB and MC areandrespectively. What is Jerry's best choice?
Answer:
Using the Principle of No Marginal Improvement you can solve for Jerry's best choice by setting MB = MC and solving for S:
=
So, Jerry's best choice is to hire a personal trainer for 12.67 (twelve and two-thirds of a session) sessions.
4. Using the information from (3), what happens if the gym decides to charge Jerry a one-time non-refundable service fee of $25 for the use of a new member-trainer work-out area. How does this fee impact Jerry's choice of the best action in (3) and why?
Answer:
Introducing a one-time non-refundable consultation fee of $25 would have the effect of changing your total cost function from, to. On the graph from (8) the TC curve would be shifted up by 25 units. Despite this shift in the TC curve this does not change the marginal cost curve. Since the MC function and the MB function are the same you will find that your best choice from (8) is the same, even after adding a one-time service fee. The service fee is what is known as a sunk cost. Sunk costs have no effect on Jerry's best choice because regardless of how many sessions he employs his trainer, the service fee is constant.
5. What do you think might happen to Jerry's cost schedule if the gym charged a new fee of $5 per session for using the area? Do you think there might be a new best choice?
Answer:
If the gym introduced a new $5 per session fee for the new member-trainer area Jerry's cost schedule would be changed. The cost for each of Jerry's choices would increase by 5 times the number of total sessions. It is also likely that Jerry's benefit schedule could see and increase due to the new training area.
It is possible that if this new service fee was implemented that Jerry would be looking at a new best choice depending on how much the benefits outweigh the costs at each choice.
Additional Exercises for Chapter 3
. Assume you have a few hours free on a Saturday afternoon and you've ranked your options, starting with your most preferred, as (1) going to see a movie, (2) going to the gym, (3) reading a book or (4) watching grass grow. What is your opportunity cost of going to see a movie and why?
2. Why is it important to make a decision based on the net benefit of your options and not just the total benefit?
3. Suppose that you hire a mechanic to fix your motorcycle for up to six hours. The total benefit and total cost of the repair work are in the table below. Calculate the net benefit for each choice and use your answer to identify the best option.
Hours
Benefit
Cost
Net Benefit
0
0
0
460
80
2
840
220
3
140
420
4
360
680
5
500
000
6
560
380
4. Using the information from (3), what happens if the mechanic decides to impose an additional $50 hourly fee? What is the maximum net benefit now?
5. Calculate the marginal benefit and marginal cost to fill in the table below. How can you use your answers to identify the best choice?
Hours
Benefit
Marginal Benefit
Cost
Marginal Cost
0
0
0
460
80
2
840
220
3
140
420
4
360
680
5
500
000
6
560
380
6. Suppose that you hire a mechanic to fix your motorcycle for up to six hours. The total benefit of the repair work is and the total cost is. The corresponding functions for marginal benefit and marginal cost are and, respectively. What is your best choice?
7. The figure below displays the functions for total benefit and total cost for a decision. How can you use this information to determine the best choice? What is the maximum net benefit?
8. Suppose again that you hire a mechanic to fix your motorcycle for up to six hours. The total benefit of the repair work is and the total cost is. The corresponding functions for marginal benefit and marginal cost are and, respectively. What is your best choice?
9. Suppose that you hire a mechanic to fix your motorcycle for up to six hours. The total benefit and total cost of the repair work are in the table below. Calculate the net benefit for each choice and use your answer to identify the best option.
Hours
Benefit
Cost
Net Benefit
0
0
0
335
65
2
640
80
3
915
345
4
160
560
5
375
825
6
560
140
0. Using the information from (8), what happens if the mechanic decides to charge a one-time non-refundable consultation fee of $100? How does this fee impact your choice of the best option in question (8) and why?
1. Why should sunk costs be ignored when finding the best choice from a set of options?
2. How can the No Marginal Improvement Principle be used to find the best possible choice over a set of options?
3. Suppose once more that you hire a mechanic to fix your motorcycle for up to six hours. The total benefit of the repair work is and the total cost is. The corresponding functions for marginal benefit and marginal cost are and, respectively. What is your best choice? What if the mechanic requires a 1-hour minimum payment?
4. What is the difference between an interior action and a boundary action and why is it important to eliminate the boundary choices?
5. Suppose that you find the largest net benefit for repairs to be 4.3 hours but the mechanic requires that payments be made in one-hour increments, which forces you to choose either 4 hours or 5 hours. Which choice do you make and why?
Chapter 4
Sample Problems
. Jimmy is going to fraternity initiation dinner and has ranked his preference bundles for rolls of sushi and number of enchiladas according to the table below. Using the information on the table below, do Jimmy's rankings show support for the More-Is-Better Principle? Also, using the information on the table, can you tell whether Jimmy prefers sushi to enchiladas, or enchiladas to sushi?
Sushi
(rolls)
3
1
6
3
2
3
8
4
2
4
9
7
5
0
5
4
2
0
0
2
3
Enchiladas
(Number of enchiladas)
Answer:
To check whether Jimmy's preferences satisfy the more-is-better principle, notice that in any particular column the numbers at the top are smaller than the numbers at the bottom. Similarly, in any particular row, the numbers on the right-hand side are smaller than the numbers on the left-hand side. These two aspects of Jimmy's preferences imply that if either the number of rolls of sushi (number of enchiladas) is held constant Jimmy prefers more enchiladas (sushi).
2. Using the information from question (1) and starting with 2 rolls of sushi and 1 enchilada, will Jimmy
a) Swap 1 sushi roll for 1 more enchilada?
b) Swap 2 sushi rolls for 2 more enchiladas?
c) Swap 1 enchilada for 1 more sushi roll?
Answer:
a) Yes, he would swap
b) No, he will not swap
c) No, he will not swap
3. The figure below contains two indifference curves one for Jimmy and another curve for Ron, Jimmy's new fraternity brother. The points represent their current consumption bundles. Using the graphs, is trade favorable between the two? Explain your answer.
Answer:
Jimmy's current consumption bundle has him on the steep part of his indifference curve where he is willing to forgo more sushi in return for some more enchiladas. In contrast, Ron's current consumption bundle has him on the flat part of his indifference curve where he is willing to forgo more enchiladas in return for some more sushi. Therefore, a trade in which Jimmy gives Ron some of his sushi in return for a greater number of enchiladas than was previously possible would benefit both.
4. At their current consumption bundles, who has a greater marginal rate of substitution for enchiladas with sushi, Jimmy or Ron? Use the graph in (3). Explain your answer.
Answer:
According to their indifference curves, enchiladas are worth more to Jimmy than to Ron. This is evident from the slopes of both of their indifference curves. Jimmy is less willing to forgo enchiladas for more sushi than Ron. Thus, Jimmy's MRS for enchiladas with sushi is higher than Ron's.
5. Ron's preferences for sushi (S) and enchiladas (E) in a month correspond to the utility function How does Ron rank the following alternatives?
a) 1 roll of sushi and 4 enchiladas.
b) 4 rolls of sushi and 1 enchilada.
c) 2 rolls of sushi and 2 enchiladas.
d) 3 rolls of sushi and 2 enchiladas.
Answer:
Use Ron's utility function to find to help find the order of preference between the alternatives.
For example for (a): The rest are as follows: Therefore, Ron's preferences are as follows: 1st, 4 rolls of sushi and one enchilada; 2nd, 3 rolls of sushi and 2 enchiladas; and lastly, both (a) one roll of sushi and (c) 4 enchiladas, and 2 rolls of sushi and 2 enchiladas.
6. Given Ron's utility function in question (5), find a formula for his indifference curves. What is Ron's marginal utility of sushi? What is Ron's marginal utility of enchiladas? What is her marginal rate of substitution for sushi with enchiladas?
Answer:
Given Ron's utility function, the formula his indifference curves can be found by rearranging the equation and solving for E, as such:
(1)
Ron's marginal utility of sushi is (addingincreases the utility value by 3 so,)
The marginal utility of enchiladas is (adding ?E the utility value by so,
Now, to find Ron's marginal rate of substitution for sushi with enchiladas, use the equations for finding the MRS with the information about the marginal utilities of both sushi and enchiladas, as such:
Her MRS of sushi with enchiladas is enchiladas per roll of sushi.
Additional Exercises for Chapter 4
. Why is it important that consumers follow the Choice Principle? What would happen if they didn't?
2. Stephanie is going to a birthday party and has ranked her preference bundles of pieces of pie and scoops of ice cream according to the table below. Using the information in the table, how do Stephanie's rankings show support for the More-is-Better Principle?
Pie (pieces)
3
0
5
2
2
3
7
4
3
4
9
8
6
0
5
4
2
1
0
2
3
Ice Cream
(scoops)
3. Using the information in question (2) and starting with 1 piece of pie and 1 scoop of ice cream, will Stephanie
a) swap 1 scoop of ice cream for 1 piece of pie?
b) swap 1 scoop of ice cream for 2 pieces of pie?
c) swap 1 piece of pie for 2 scoops of ice cream?
4. Does the Ranking Principle allow for ties? What does this mean about Stephanie's preferences for pie and ice cream in question (2)?
5. The figure below contains two indifference curves and several options for consumption bundles for a consumer. Using this information, rank the bundles from most preferred to least preferred according to this consumer's preferences.
6. Suppose that Stephanie's preferences for pieces of pie (P) and scoops of ice cream (IC) follow the formula. Plot three indifference curves for Stephanie when , , and when on the graph.
7. Now suppose that Stephanie will only eat pie and ice cream in a 1-to-1 ratio and gains no additional benefit from any other combination. Draw a few examples of this indifference curve on the graph. What is this type of preference called?
8. Now suppose that Stephanie is asked about her preferences regarding pie and cake. She prefers pie and cake equally, but doesn't like to consumer them together. If pie and cake are perfect substitutes for Stephanie, draw her indifference curves if she has only 1 serving, 2 servings, or 3 servings of either pie or cake.
9. Stephanie has brought her daughters Ashleigh and Taylor to the birthday party. Ashleigh loves cake but hates pie, while Taylor loves pie but hates cake. Draw Ashleigh and Taylor's indifference curves on the graph if both girls each have 2 servings of their choice.
0. Stephanie's preferences for pie (P) and ice cream (IC) in a month correspond to the utility function. How does Stephanie rank the following alternatives:
a) 1 piece of pie and 4 scoops of ice cream
b) no pie and 5 scoops of ice cream
c) 4 pieces of pie and no ice cream
d) 2 pieces of pie and 3 scoops of ice cream
1. Given Stephanie's utility function in question (10), find a formula for her indifference curves. What is Stephanie's marginal utility of pie? What is Stephanie's marginal utility of ice cream? What is her marginal rate of substitution for pie with ice cream?
2. Given Stephanie's utility function in question (10), plot her indifference curves for, , and on the graph below.
3. Stephanie's husband Dave also likes pie and ice cream, but his utility function takes the form. Find a formula for Dave's indifference curves. What is Dave's marginal utility of pie? What is Dave's marginal utility of ice cream? What is his marginal rate of substitution for pie with ice cream?
4. If Dave's utility function is as in question (13) and he currently has 1 piece of pie and 2 scoops of ice cream, would he be willing to trade his current bundle for 2 pieces of pie and 1 scoop of ice cream? Why or why not?
5. Suppose Stephanie and Dave have preferences according to their utility functions in questions (10) and (13), respectively. If Stephanie has 4 scoops of ice cream and Dave has 3 pieces of pie, is there a trade they could make to improve both of their positions? Explain your answer.
Chapter 5
Sample Problems
. In the Chapter 4 exercises we examined Jimmy's preferences regarding rolls of sushi and numbers of enchiladas. If a roll of sushi is $2.50 and an enchilada is $1.50, which bundles can Jimmy afford if his income is $7.00? Fill in the table below with the total cost of each bundle and identify his affordable options. What happens to the number of affordable options if his income increases to $9.00?
Sushi
(rolls)
3
2
0
0
2
3
Enchiladas
(Number of enchiladas)
Answer:
Sushi
(rolls)
3
7.5
9
0.5
2
2
5
6.5
8
9.5
2.5
4
5.5
7
0
0
.5
3
4.5
0
2
3
Enchiladas
(Number of enchiladas)
If Jimmy's income is $7.00, his affordable options are lightly shaded.
If Jimmy's income is $9.00, his affordable options increase to include all of the lightly shaded options plus the new options which are more darkly shaded.
2. Suppose a sushi roll (S) costs $1.50 and an enchilada (E) costs $2.00, first plot Jimmy's initial budget line on the graph below when his income equals $6.00. Now suppose the price of sushi rolls doubles to $3.00. Plot Jimmy's new budget line and calculate the horizontal intercept, the vertical intercept, and the slope of his budget line. What happened to the number of affordable options and why?
Answer:
If the price of sushi doubles to $3.00 the new budget line is. His new budget line has a slope of, with a horizontal intercept at 3 enchiladas, and a vertical intercept at 2 rolls of sushi. The number of affordable options has decreased due to the increase in the price of rolls of sushi. This increase in the price of sushi has caused the original budget line to rotate inward from the origin (pivoting at the intercept for enchiladas).
3. Taking her friends advice before the start of the next semester, Amy decides to allocate $24 dollars per month of her income for the consumption of coffee and energy drinks instead of $40 per month. Coffee (C) costs $6 per pound and energy drinks (E) cost $12 per package. Fractions are allowed since the energy drink packages can be split and Amy can purchase coffee in uneven increments. If Amy'ss marginal rate of substitution is, how many pounds of coffee and energy drink packages will she buy?
Answer:
First, derive the equation for the budget line as follows:
.
(Budget line)
Now, use the tangency condition to help solve for C:
Next, use the previous result along with the equation for the budget line to solve for C and E:
&
Now use E to find C.
If Amy'ss marginal rate of substitutions for coffee with energy drinks is, then she should buy 3 pounds of coffee andpackage of energy drinks.
4. As in question (3), Amy spends money on coffee and energy drinks for the spring semester. Coffee (C) costs $6 per pound and energy drinks cost $12 per package. Suppose Amy's preferences correspond to the utility function. For that utility function, the marginal utility of coffee is 3E (the amount of coffee) and the marginal utility of energy drinks is (the number of energy drink packages). Fractions are still acceptable. Using,, and, draw Amy's income-consumption curve and her Engel curves for coffee and energy drinks.
Answer:
To plot the Income-consumption curves we first need to find the budget lines. Since the prices of coffee and energy drinks are constant and the only thing changing is income it is fairly simple to find the budget lines:
If:()
If:()
If:()
Next, we need to find the optimal bundles for each budget line. To do this we use the following equation:
Using the previous result you can solve for the optimal bundles at each budget line. The optimal bundles are as follows:
For:, For:, For:
Now we have all of the information needed to plot the income-consumption curves and the Engel curves.
Additional Exercises for Chapter 5
. In the Chapter 4 exercises we examined Stephanie's preferences regarding pieces of pie and scoops of ice cream. If a piece of pie is $2.00 and a scoop of ice cream is $1.00, which bundles can Stephanie afford if her income is $4.00? Fill in the table below with the total cost of each bundle and identify her affordable options. What happens to the number of affordable options if her income increases to $6.00?
Pie (pieces)
3
2
0
0
2
3
Ice Cream
(scoops)
2. Using the information in question (1), plot Stephanie's budget lines on the graph below when her income is $4.00. Calculate the horizontal intercept, the vertical intercept, and the slope of her budget line. What happens to her budget line if her income increases to $6.00? Again, calculate the horizontal intercept, the vertical intercept, and the slope of her budget line.
3. Using the information in question (1), first plot Stephanie's initial budget line on the graph below when her income equals $4.00. Now suppose the price of ice cream doubles to $2.00. Plot Stephanie's new budget line and calculate the horizontal intercept, the vertical intercept, and the slope of her budget line. What happened to the number of affordable options and why?
4. Using the information in question (1), first plot Stephanie's initial budget line on the graph below when her income equals $4.00. Now suppose that pie goes on sale for $1. Plot Stephanie's new budget line and calculate the horizontal intercept, the vertical intercept, and the slope of her budget line. What happened to the number of affordable options and why?
5. The table below reproduced from the Chapter 4 exercises shows Stephanie's preferences for pie and ice cream. First use your answer for question (1) when her income is $4.00 to shade in the unaffordable bundles. Next identify which bundle would maximize Stephanie's utility.
Pie (pieces)
3
0
5
2
2
2
7
4
3
4
9
8
6
0
5
4
3
1
0
2
3
Ice Cream
(scoops)
6. The table below reproduced from the Chapter 4 exercises shows Stephanie's preferences for pie and ice cream. First use your answer for question (1) when her income is $6.00 to shade in the unaffordable bundles. Next identify which bundle would maximize Stephanie's utility. How does this choice compare to your answer for question (5)?
Pie (pieces)
3
0
5
2
2
2
7
4
3
4
9
8
6
0
5
4
3
1
0
2
3
Ice Cream
(scoops)
7. Continuing with the example, what would happen if pie was rationed to a maximum of 2 pieces per person? Using the budget line below when the price of pie is $2.00, the price of ice cream is $1.00, and Stephanie's income is $8.00, demonstrate the impact of this rationing. Using her preferences given in question (5), how does pie rationing impact Stephanie's decision?
8. Now suppose that Stephanie's preferences are represented by three ranked indifference curves in the figure below. If pie and ice cream are $2.00 each and Stephanie's income is $8.00, which bundle will she choose? Plot her budget line and identify which indifference curve achieves this utility-maximizing bundle.
9. Why is it important that the budget line be tangent to the indifference curve at a utility-maximizing bundle for interior solutions? Why are boundary solutions excluded from the tangency condition?
0. Holly teaches second grade and has a budget of $50 per month to spend on candy and balloons for class parties. Candy (C) costs $5 per pound and balloons cost $10 per package. Fractions are allowed since the balloon packages can be split and Holly can purchase candy in uneven increments. If Holly's marginal rate of substitution is, how many pounds of candy and balloon packages will she buy?
1. Using the information in question (10), how many pounds of candy and balloon packages will Holly buy if her marginal rate of substitution is?
2. As before, Holly is buying candy and balloons for her class. Her preferences correspond to the utility function. For that utility function, the marginal benefit of candy is B (the number of balloon packages) and the marginal benefit of balloons is C (the amount of candy). Fractions are still acceptable. If candy (C) costs $5 per pound and balloons cost $10 per package and Holly's budget is $50, how many of each good will she buy for her classroom?
3. Using the information in question (12), calculate the utility-maximizing amounts of candy and balloons if Holly's utility function is . For that utility function, the marginal benefit of candy is and the marginal benefit of balloons is . If candy (C) costs $5 per pound and balloons cost $10 per package and Holly's budget is $50, how many of each good will she buy for her classroom?
4. As in question (10), Holly's marginal rate of substitution for candy and balloons is. Holly's budget is $50 and the price of candy is $5. Calculate her optimal amounts of candy and balloons when the price of balloons is $5 and when the price of balloons is $10. Draw her price-consumption curve and her demand curve for balloons on the graphs below.
5. Assume Holly's marginal rate of substitution for candy and balloons is . Holly's budget is $50 and the price of candy is $5. Calculate her optimal amounts of candy and balloons when the price of balloons is $5 and when the price of balloons is $10. Draw her price-consumption curve and her demand curve for balloons on the graphs below.
6. As in question (10), Holly spends money on candy and balloons for class parties. Candy (C) costs $5 per pound and balloons cost $10 per package. Suppose Holly's preferences correspond to the utility function. For that utility function, the marginal benefit of candy is B (the number of balloon packages) and the marginal benefit of balloons is C (the amount of candy). Fractions are still acceptable. Using , , and , draw Holly's income-consumption curve and her Engel curves for candy and balloons.
7. As in question (16), Holly spends money on candy and balloons for class parties. Candy (C) costs $5 per pound and balloons cost $10 per package. Suppose Holly's preferences correspond to the utility function. For that utility function, the marginal benefit of candy is and the marginal benefit of balloons is . Fractions are still acceptable. Using , , and , draw Holly's income-consumption curve and her Engel curves for candy and balloons.
8. How can the income-consumption curve be used to identify whether a good is normal or inferior?
Chapter 6
Sample Problems
. Starting a new semester, Amy has budgeted $24 per month to spend of coffee (C) and energy drinks (E). Coffee costs $6 per pound and energy drinks cost $12 per pack. Fractions are allowed as energy drink packs may be split and coffee may be purchased in uneven increments. Amy marginal rate of substitution isand the formula for her indifference curve is Suppose the price of energy drinks falls from $12 to $6. What is the uncompensated effect of her purchases of coffee and energy drinks? What is the compensated effect? How much compensation is involved? Decompose the uncompensated price change into the substitution and income effects.
Answer:
To find the uncompensated effect of Amy's purchases of coffee and energy drinks first use the tangency condition to solve for the E and C using a generic form of a budget constraint as follows:
(Budget line)
Tangency condition:
Now use the previous two results to solve for E and C:
&
So, withshe chooses. If the price of energy drinks falls to $6.00 she chooses.This uncompensated price reduction raises Amy's energy drink package purchases from to 1.5 and decreases the amount of coffee she buys from 8 to 5.
Next, to find the compensated effect we need to used the tangency condition and find the relationship between coffee and energy drinks at their new prices.
If, the relationship between candy and balloons is:
Amy's best choice with a compensated price change also lies on the indifference curve that passes through the bundle. Using the formula for her indifference curvethe value of U that runs through the bundleis 12.5, which implies that Holly's formula for this indifference curve is:
A bundle can satisfy both the tangency condition formula and the indifference curve formula only if: Solving for C using the quadratic equation. Using the tangency condition, we get that. So the compensated price effect shifts Holly from the bundleto the bundle.
At the new prices, Amy needs $30.42 to buy the bundle. Since her income is $24, we would have to give Amy $6.84 to compensate for the price cut.
To break up the uncompensated price change into substitution effects and income effects. Note that the substitution effect is the same as the effect of the compensated price change. The income effect is what is known as the residual; it shifts Amy from the bundleto the bundle.
2. Assume that Amy has a budget of $24 a month to spend on coffee (C) and energy drinks (E). Coffee cost $6 per pound and energy drinks cost $12 per package. Amy's marginal rate of substitution isand the formula for her indifference curve is After being recommended by her doctor to stop drinking coffee for one month, Amy was left with only energy drinks to keep her up during late night study sessions and early morning classes. Her friends know how much Amy likes coffee so they would like to help her through the month. Calculate her compensating variation.
Answer:
First we must use the budget line and the tangency condition to solve for optimal values of coffee and energy drinks.
(Budget line)
Tangency condition:
Solving for C and E we get&
Next, use the utility function in its functional form to solve for U with the optimal bundle:
Amy's indifference curve formula through this bundle is:
If the Amy is not allowed to have any coffee, thenTherefore, to place Amy on the same indifference curve, we must pick the value of E for whichSolving for E, we getTo buy 5 packages of energy drinks Amy would need $60.00, so her friends would have to lend her $36.00 to compensate for not having coffee for one month. This is Amy's compensation variation.
3. Suppose that Eric is buying coffee and energy drinks for the last month of the semester. His preferences correspond to the utility function.When coffee costs $6 per pound and energy drinks cost $12 per package and Eric's budget is $24, he buys 2 pounds of coffee and 1 package of energy drinks. Find the compensated demand curve for energy drinks that passes through this point and graph it.
Answer:
To find the equation of the compensated demand curve first take the formula for the tangency condition and set the price of coffee equal to 10:
Next, rewrite the utility function as follows using the bundle given:
Now set the tangency condition formula equal to the indifference curve formula equal to each other and solve for C:
This is the compensated demand curve.
4. Tom is a college senior that wants to analyze the change in cost of living over the four years he spent in college. He tracked his purchases and the change in prices of a bundle consisting of coffee, cereal, and pizza.
2004
2005
2006
2007
Coffee
97 cups at $1.65
20 cups at $1.70
55 cups at $1.70
90 cups at $1.75
Cereal
45 boxes at $2.45
35 boxes at $2.60
43 boxes at $2.70
50 boxes at $2.75
Pizza
50 pizzas at $7.00
40 pizzas at $7.50
35 pizzas at $7.80
25 pizzas at $8.00
a) Using 2004 as the base period, create a Laspeyres price index to determine how Tom's cost of living changed. How did his cost of living change according to this measure?
b) Repeat part (a) first using 2005 as the base period and then using 2006 as the base period.
c) Do the price indexes you found in parts (a) and (b) imply an increase in Tom's cost of living over time? If not, by how much do they differ?
Answer:
a) 2004 (base period)- use 2004 quantity and prices
2005- use 2004 quantity and 2005 prices
2006- use 2004 quantity and 2006 prices
2007- use 2004 quantity and 2007 prices
Lasperyres index using 2004 as base year
2004:
2005:
2006:
2007:
b) 2005 Base Period
2004- use 2005 quantity and 2004 prices
2005 (base period)- use 2005 quantity and 2005 prices
2006- use 2005 quantity and 2006 prices
2007- use 2005 quantity and 2007 prices
Lasperyres index using 2005 as base year
2004:
2005:
2006:
2007:
2006 Base Period
2004- use 2006 quantity and 2004 prices
2005- use 2006 quantity and 2005 prices
2006 (base period)- use 2006 quantity and 2006 prices
2007- use 2006 quantity and 2007 prices
Lasperyres index using 2006 as base year
2004:
2005:
2006:
2007:
c) No, the indexes found in (a) and (b) do not imply the same percentage increases in Tom's cost-of-living over time. If 2004 is treated as the base year, then the percentage change in the cost-of-living from 2004 to 2007 was 11.8%. If 2005 was treated as the base year, then the percentage change increase was 10.5%. Finally, for base year 2006, the percentage change increase was 9.72%. So between the different base years they differed by a matter of about 2.8%.
Additional Exercises for Chapter 6
. What is the difference between the substitution effect of a price change and the income effect of a price change? How do they relate to an uncompensated price change?
2. Holly teaches second grade and has a budget of $50 per month to spend on candy and balloons for class parties. Candy (C) costs $5 per pound and balloons cost $10 per package. Fractions are allowed since the balloon packages can be split and Holly can purchase candy in uneven increments. Holly's marginal rate of substitution is and the formula for her indifference curve is . Suppose the price of balloons falls from $10 to $5. What is the uncompensated effect of her purchases of candy and balloons? What is the compensated effect? How much compensation is involved? Decompose the uncompensated price change into the substitution and income effects.
3. Holly's options and budget are as in question (2), except her marginal rate of substitution is now and the formula for her indifference curve is . Suppose the price of balloons falls from $10 to $5. What is the uncompensated effect of her purchases of candy and balloons? What is the compensated effect? How much compensation is involved? Decompose the uncompensated price change into the substitution and income effects.
4. The graph below shows the impact of an increase in the price of good X. Demonstrate the income effect by drawing the compensated budget line. Identify the substitution effect by labeling the bundle chosen before the price change, the bundle chosen after the price change, and the bundle chosen on the compensated budget line.
5. The graph below shows the impact of a decrease in the price of good Y. Demonstrate the income effect by drawing the compensated budget line. Identify the substitution effect by labeling the bundle chosen before the price change, the bundle chosen after the price change, and the bundle chosen on the compensated budget line.
6. Explain why the income effect is negative for a price increase if a good is normal and positive for a price increase if a good is inferior.
7. What is a Giffen good and how could a good with these characteristics violate the Law of Demand?
8. Assume that everything is as in question (2) and Holly has a budget of $50 per month to spend on candy and balloons for class parties. Candy (C) costs $5 per pound and balloons cost $10 per package. Holly's marginal rate of substitution is and the formula for her indifference curve is . When Holly gets to the store she finds that they are out of balloons. Since Holly is a loyal customer, the store owner would like to know how much it would cost to make up for this issue. Calculate the compensating variation.
9. Assume that everything is as in question (2) and Holly has a budget of $50 per month to spend on candy and balloons for class parties. Candy (C) costs $5 per pound and balloons cost $10 per package. Holly's marginal rate of substitution isand the formula for her indifference curve is . When Holly gets to the store she finds that they are out of balloons. Since Holly is a loyal customer, the store owner would like to know how much it would cost to make up for this issue. Calculate the compensating variation.
0. The formula for Holly's monthly demand curve for pencils is . Suppose that pencils cost $1.00 each. First graph the demand curve and shade in the area representing Holly's consumer surplus. Calculate her consumer surplus. What if the price of pencils falls to $0.50? Calculate the change in consumer surplus and show this change in surplus on the graph.
1. The formula for Holly's monthly demand curve for bottles of paint is . Suppose that bottle of paint cost $6.00 each. First graph the demand curve and shade in the area representing Holly's consumer surplus. Calculate her consumer surplus. What if the price of paint increases to $7? Calculate the change in consumer surplus and show this change in surplus on the graph.
2. What is a cost-of-living index? How does the Laspeyres price index possibly overstate the increase in the cost of living?
3. Explain how an increase in the wage rate could either increase or decrease the quantity of labor supplied. When would each of these responses be likely to occur?
4. What is the difference between an uncompensated and a compensated demand curve? When would they be the same and when would they differ?
5. As before, Holly is buying candy and balloons for her class. Her preferences correspond to the utility function.When candy costs $5 per pound and balloons cost $10 per package and Holly's budget is $50, she buys 5 pounds of candy and 2.5 packages of balloons. Find the compensated demand curve for candy that passes through this point and graph it.
6. As before, Holly is buying candy and balloons for her class. Her preferences correspond to the utility function.When candy costs $5 per pound and balloons cost $10 per package and Holly's budget is $50, she buys 7 pounds of candy and 1.5 packages of balloons. Find the compensated demand curve for candy that passes through this point and graph it.
Chapter 7
Sample Problems
. The family of isoquants for a firm hiring interns with a bachelor's degree in Economics and interns with a bachelor's in Business Administration appears in the graph below. What does this graph demonstrate about the substitutability of the inputs for this firm? Explain.
Answer:
The graph above shows that for this firm Economics majors are perfect substitutes for Business majors. The firm is willing to substitute one Economics major for 3 business majors.
2. Alex and Jessica own a coffee shop close to the local college campus. Their long run production function is the marginal product of labor is and the marginal product of capital is . Calculate the marginal rate of technical substitution for labor with capital, . Does their technology have increasing, decreasing, or constants returns to scale?
Answer:
Since Alex and Jessica's production function is a Cobb-Douglas production function of the form finding whether their technology has increasing, decreasing, or constant returns to scale becomes a matter of checking the sum of the exponents and .
Therefore Alex and Jessica's technology exhibits increasing returns to scale.
3. Suppose Alex and Jessica close their coffee shop two hours early for one whole week to train their baristas to make better coffee at a faster rate, so their production of labor doubles relative to their production of capital. What happens to the shop's MRTS? Is this a factor-neutral technical change? Explain.
Answer:
The MRTS changes in its slope, because labor is more productive relative to capital. This cannot be a factor-neutral technical change because the MRTS changed due to the improvement of labor production.
4. Alex and Jessica own two coffee shops that supply coffee to the local university. Labor is their only variable input. Suppose the marginal product of labor in Shop A is , where is the number of workers allocated to Shop A. The marginal product of labor of Shop B is , where is the number of laborers assigned to Shop B. Suppose that Alex and Jessica employ a total of 10 workers. What is the best assignment of the 10 workers to the two shops?
Answer:
First, write the as a function of the number of laborers assigned to Shop A, LA:
Next, find the level of LA, that equates the MPL in the two shops. This can be done by setting
So Alex and Jessica should assign 6 workers to Shop A, and the other 4 workers to Shop B.
Additional Exercises for Chapter 7
. Giselle owns a specialty lemonade stand called Pucker Up. Her production options for several amounts of labor appear in the table below. First, identify which combinations are efficient and which are inefficient. What characteristic determines whether a production method is inefficient?
Production
Method
Units of Labor
Output
Efficient?
A
48
B
2
24
C
2
12
D
3
205
E
3
216
F
4
312
G
4
295
2. How is a firm's efficient production frontier from its production possibilities set? Why should a firm aim to achieve a point on its efficient production frontier?
3. Giselle can produce her specialty lemonade at Pucker Up daily according to the production function . How ...
This is a preview of the whole essay
Production
Method
Units of Labor
Output
Efficient?
A
48
B
2
24
C
2
12
D
3
205
E
3
216
F
4
312
G
4
295
2. How is a firm's efficient production frontier from its production possibilities set? Why should a firm aim to achieve a point on its efficient production frontier?
3. Giselle can produce her specialty lemonade at Pucker Up daily according to the production function . How many cups of lemonade can Giselle produce daily with one worker? What happens when she adds a second unit of labor? What do your answers show about the marginal product of labor at low levels of inputs?
4. Assume Giselle can produce her specialty lemonade according to the production function . How many cups of lemonade can Giselle produce daily with one worker? What happens when she adds a second unit of labor? What do your answers show about the marginal product of labor at low levels of inputs?
5. Giselle can produce lemonade at Pucker Up according to the production function . Graph Giselle's production function for up to 5 workers, supposing that the number of workers is finely divisible (meaning Giselle can hire workers part-time, full-time, etc.) Shade in Giselle's production possibilities set and identify her efficient production frontier.
6. Given Giselle's production function in question (5), where , what is the maximum amount of labor that Giselle would choose to hire? Why would she not hire any additional workers beyond this level?
7. Giselle can produce lemonade at Pucker Up according to the production function . Graph Giselle's production function for up to 5 workers, supposing that the number of workers is finely divisible (meaning Giselle can hire workers part-time, full-time, etc.) Shade in Giselle's production possibilities set and identify her efficient production frontier.
8. What is an isoquant and why is it that an isoquant must not slope upward?
9. Use Giselle's production function [] for Pucker Up to fill in the table below. Calculate the output, the marginal product of labor () and the average product of labor () for each level of labor input. What do you notice about the as the number of workers increases?
Units of Labor
Output
0
2
3
4
5
6
0. Use Giselle's production function [] for Pucker Up to graph the marginal product of labor () and the average product of labor () for each level of labor input. How are the marginal product of labor and the average product of labor related?
1. Suppose that Giselle is making a long run decision and can now vary both the quantity of labor, L, and her capital, K, which is the number of juicing machines she uses at Pucker Up. Her new production function is . Fill in the table below for a given number of workers and number of juicers. Using your answers, what can you say about the productivity of labor and capital at Pucker Up?
Units of Labor
Units of Capital
Output
2
2
2
2
2
3
3
2
3
3
2. For Giselle's long run production function the marginal product of labor isand the marginal product of capital is . Calculate the marginal rate of technical substitution for labor with capital, . Would the isoquant exhibit a declining marginal rate of substitution in this case? Why or why not?
3. The family of isoquants for a firm producing pre-made peanut butter and jelly sandwiches appears in the graph below. What does this graph demonstrate about the substitutability of inputs for this firm? Explain.
4. Giselle can produce her specialty lemonade according to the production function . Does her technology have increasing, decreasing, or constant returns to scale? What if instead her production function was ? Compare your answers. Why are they the same or why do they differ?
5. Now suppose Giselle moves Pucker Up to a new location and increases her productivity to . What happened to her marginal rate of technical substitution after the improvement? What is this type of productivity improvement called?
Chapter 8
Sample Problems
. Alex and Jessica own a coffee shop near campus. Suppose that their daily production function for workers, L, and capital K, is The wage rate is $100 daily and a unit of capital costs $50 daily. What is Alex and Jessica's least-cost input combination to make 145 cups of coffee per day? What is their total cost?
Answer:
We know that thefor the Cobb-Douglas production functionis , which is a declining MRTS.
To locate an interior point combination, set the equal to the input price ratio:
and substitute the values for,,,. Thus,
When Alex and Jessica use L units of labor and 6L units of capital, output is
So to produce 145 cups of coffee per day requiresworkers (about 5 workers, L = 5). Since, they need 30 units of capital. Their total cost is, $2000 per day.
2. Using the information given in question (1) derive the cost function?
Answer:
In problem (1) we found that to produce Q cups of coffee Alex and Jessica need the amount of labor L that solves the formula:and she needs 6 times the amount of capital. Thus, Alex and Jessica needworkers andunits of capital. The cost of producing Q units is therefore,
3. Suppose Alex and Jessica decided to open a new coffee shop to service the local business center. Their production costs are at the first shop and at the second shop, where and are the number of cups of coffee produced daily at each shop. The corresponding marginal costs are and. If Alex and Jessica want to produce 400 cups of coffee per day efficiently, how many cups should each shop produce? What is their total cost of production?
Answer:
We know that with a least-cost plan that assigns a positive amount of output to each shop, marginal cost must be the same at both locations. Thus Alex and Jessica need to divide production between their two stands so that their marginal costs are equal. Doing this means choosingandso that MC1=MC2. Since total output equals 400 cups of coffee, we knowso we can express MC2 as follows:
Next, to find how much each stand will produce, set MC1=MC2:
Therefore, Alex and Jessica must produce 300 cups of coffee in shop 1 and 100 cups in shop 2. Their total cost of production will be:
4. Suppose a firm faces the cost function. What is the fixed cost F? What is the variable cost,? Derive the average cost (AC), the average variable cost (AVC) and the average fixed cost (AFC).
Answer:
The fixed cost for the firm is $250, since it is the only part of the cost function that does not depend on Q. The variable cost,, in the function above is , because these are the parts of the cost function that vary with changes in Q.
To find the average cost (AC) use the fact that. In this firm's case it is:
and simplifying:
To find the AVC and AFC use the following equation:
Now, . Thus, and
5. Suppose Alex and Jessica decide to make blended ice drinksas well as coffeeat their shops. Their new cost function is. Does Alex's and Jessica's firm exhibit economies of scope at the production bundle, if the cost functions for two competing firms in the industry specializing in either coffee or blended ice drinks are (Firm 1) and (Firm 2)?
Answer:
Economies of scope would occur in this example if Alex's and Jessica's firm produced both coffee and blended ice drinks more cheaply than the other two firms. In other words, for the bundle, .
In this example evaluate as follows:
Alex's and Jessica's firm,
For the other two firms,
Thus Alex's and Jessica's firm produces the bundle more cheaply than both firm 1 and firm 2 combined, demonstrating economies of scope.
Additional Exercises for Chapter 8
. What characteristic distinguishes a sunk fixed cost from an avoidable fixed cost? Why is it important to understand the difference?
2. Giselle owns a specialty lemonade stand called Pucker Up. Her short-run production function is . Giselle faces costs of $10 per hour in wages and sunk costs of $75 to lease her stand. What is her short-run cost function for producing cups of lemonade?
3. Giselle's lemonade stand has a short-run production function is . Giselle faces costs of $10 per hour in wages and sunk costs of $75 to lease her stand. What is her short-run cost function for producing cups of lemonade?
4. Assume Giselle can vary both the quantity of labor, L, and capital, K, which is the number of juicing machines she uses at Pucker Up. The wage rate is $80 per day and a unit of capital costs $40 per day. Using the formula, derive the isocost line for a total cost of $1200. What if ? What if ? Graph this family of isocost lines.
5. Giselle's daily production function for workers, L, and capital, K, is . The wage rate is $80 daily and a unit of capital costs $40 daily. What is Giselle's least-cost input combination to make 150 cups of lemonade per day? What is her total cost?
6. What if Giselle's daily production function for workers, L, and capital, K, is . The wage rate is $80 daily and a unit of capital costs $40 daily. What is Giselle's least-cost input combination to make 130 cups of lemonade per day? What is her total cost?
7. Using the information given in questions 5 and 6, derive the cost functions for each question.
8. Giselle is considering opening a specialty fruit punch stand called Fruityland. If she can find a space with at least 1,000 square feet, her daily production is . Additional space will not increase her production capabilities. If the wage rate is $10 per hour and the space rents for $75 per day, what is her daily cost function for producing cups of fruit punch? Graph Giselle's cost function below.
9. Giselle is considering opening a specialty fruit punch stand called Fruityland. If she can find a space with at least 1,000 square feet, her daily production is . Additional space will not increase her production capabilities. If the wage rate is $10 per hour and the space rents for $75 per day, what is her daily cost function for producing cups of fruit punch? Graph Giselle's cost function below.
0. What does it mean for a firm to produce at its efficient level of production? Does a firm always have to produce at this level of output?
1. Now Giselle has decided to expand her specialty lemonade at Pucker Up to multiple locations. Her production costs are at the first stand and at the second stand, where and are the number of cups of lemonade produced daily at each stand. The corresponding marginal costs are and . If Giselle wants to produce 225 cups of lemonade per day efficiently, how many cups should each stand produce? What is her total cost of production?
2. Again, Giselle has decided to expand her specialty lemonade at Pucker Up to multiple locations. The production costs are at her first stand and at her second stand, where and are the number of cups of lemonade produced daily at each stand. The corresponding marginal costs are and . If Giselle wants to produce 350 cups of lemonade per day efficiently, how many cups should each stand produce? What is her total cost of production?
3. Suppose a firm faces the cost function . What is the fixed cost, F? What is the variable cost, V(Q)? Derive the average cost (AC), the average variable cost (AVC) and the average fixed cost (AFC).
4. Giselle's daily production function at Pucker Up is . The wage rate is $80 daily and a unit of capital costs $40 daily. Suppose Giselle is currently using 8 juicing machines (K=8) to produce 130 cups of lemonade daily. What are Giselle's short-run and long-run cost functions?
5. Assume now that Giselle's daily production function at Pucker Up is . The wage rate is $80 daily and a unit of capital costs $40 daily. Suppose Giselle is currently using 10 juicing machines (K=10) to produce 150 cups of lemonade daily. What are Giselle's short-run and long-run cost functions?
6. Suppose a firm faces the cost function. If the firm increases its output from Q to 2Q, will the firm experience economies of scale, diseconomies of scale, or constant returns to scale? Explain.
Chapter 9
Sample Problems
. The weekly demand for Lulu's leather boots is described by the demand function . What price must Lulu charge if she wants to sell 60 pairs of leather boots per week? What is Lulu's inverse demand function?
Answer:
To determine the price at which, solve for P such that
We can find the inverse demand function by solving
for P, which gives us.
2. When a firm is a price taker, changes in its sales quantity have no effect on the price P it can change. What does this imply about a price taker's marginal revenue and marginal cost?
Answer:
When a firm is a price taker this implies that the firms marginal revenue curve is equal to the price, because the firm can only sell at the market price. Because MR = MC is a firm's profit-maximizing quantity and MR = P for a price-taking firm; it follows that P = MC.
3. Lewis owns a company that specializes in boxed wine. The market price for a box of wine is $8, and Lewis is a price taker in the market. His daily cost of producing boxed wine is , and his marginal cost is . How many boxes of wine should Lewis sell each day? What if he has an avoidable fixed cost of $650 per day?
Answer:
Use the quantity rule. Thus, Lewis' best positive sales quantity solves the formula , or
Now we must check the shut-down rule, by calculating the profit from producing 64 boxes of wine daily.
Lewis should go ahead and produce 64 boxes of wine. If Lewis also has an avoidable cost of $650 per day his profit at Q = 64 would be $522, and the shut-down rule would tell Lewis to stop producing boxed wine all together.
4. Using the information in (3), what is Lewis' supply function if he has no avoidable cost? What if his avoidable fixed cost is $650 per day?
Answer:
The best positive sales quantity, Q, is found when the price P is set equal to the marginal cost, so that
Solving for Q, we find that
If Lewis' has no avoidable fixed cost, .
Note, $4 is the minimum because is and increasing function. So using the shut-down rule, Lewis' supply function is
When we add an avoidable fixed cost of $650 per day, this does not change marginal revenue or marginal cost, but it does change when Lewis' wants to stay in business. We need to determine the new level of . This is done by setting AC = MC to determine the new efficient scales of production.
Solving for Q, we find. Substituting this quantity into the equation for average cost shows us that
So Lewis' supply function is now
5. Lewis' produces both an award-winning bottle wine, and a boxed wine, and he is a price taker. His cost function for his bottled wine is and his marginal is , whereandare his output levels of bottled and boxed wine respectively. His cost function for boxed wine isand his marginal cost function is. The price of a bottle of award-winning wine,, is $50; the price of boxed wine,, is $15. What are Lewis' profit- maximizing sales quantities?
Answer:
First, start by applying the quantity rule to find the sales quantities for both the award-winning wine and the boxed wine, where P = MC.
(1)
and, (2)
Now with equations (1) and (2) we can solve forand. This can be done in several ways but we will proceed as follows:
Rewrite (1) and (2) as (1) and (2)
Substitute (1) into (2) and solve for.
,
Solve for: ,
Lewis' profit is:
Applying the shut down rule, we have to compare this profit not only to Lewis' profit if he would shut-down his operation, but also, his profit if he would shut-down production of either of his products and only would produce one product.
If Lewis does not produce his award-winning bottled wine so that, his cost function becomesand. The best quantity of boxed wine is, which yields $2.25 profit which is less than Lewis' makes if he produces both products.
This was found by setting P = MC:
,
Lewis' profit is,
If Lewis stops producing the boxed wine so that, his cost function becomesand. Set P = MC:
Lewis' profit is,
Again, this is less than Lewis would make if he produced both boxed wine and his award-winning bottled wine, so Lewis should produce the profit-maximizing sales quantities of and.
Additional Exercises for Chapter 9
. Megan owns and operates ECON-TS4U which produces popular Economics T-shirts. Consumer demand for the T-shirts is given by. What is the inverse demand function? If Megan plans to sell 40 shirts, what price should she set? What if Megan wanted to sell 50 shirts?
2. Megan produces popular Economics T-shirts. Consumer demand for the T-shirts is given by. What is the inverse demand function? If Megan plans to sell 60 shirts, what price should she set? What if Megan wanted to sell 75 shirts?
3. Why is the demand curve horizontal for a price-taking firm?
4. Suppose that Megan also produces and sells blank T-shirts on the side. In this market, she is a price taker. Her cost of making shirts is and her marginal cost is. If the market price of a blank t-shirt is $6, how many shirts should Megan sell? Calculate Megan's profit. What if she has an avoidable fixed cost of $150?
5. Suppose that Megan is a price taker in the market for blank T-shirts. Her cost of making shirts is and her marginal cost is. If the market price of a blank t-shirt is $6.60, how many shirts should Megan sell? Calculate Megan's profit. What if she has an avoidable fixed cost of $200?
6. If Megan is a price taker in the market for blank t-shirts and her marginal cost is, what is her supply function? How would this change if she faced an avoidable fixed cost of $150?
7. If Megan is a price taker in the market for blank t-shirts and her marginal cost is, what is her supply function? How would this change if she faced an avoidable fixed cost of $200?
8. If Megan's supply function for T-shirts is, what is the minimum price at which Megan will choose to sell a positive number of units? Graph the supply function below.
9. If Megan's supply function for T-shirts is, what is the minimum price at which Megan will choose to sell a positive number of units? Graph the supply function below.
0. How does an increase in the price of a major input affect a firm's supply function? How is this effect different from that of an increase in an avoidable fixed cost? Explain.
1. If Megan's short-run and long-run cost functions are and then her short-run and long-run marginal cost functions areand. What are her short-run and long-run supply functions?
2. If Megan's short-run and long-run cost functions are and then her short-run and long-run marginal cost functions areand. What are her short-run and long-run supply functions?
3. Use the information in the graph below to calculate the producer surplus given the supply function and market price, $6. What if the firm faced a sunk cost of $100?
4. Assume the supply function for Megan's price-taking business isand the market price is $7. First, calculate Megan's producer surplus. What if Megan's business required an annual non-refundable, non-transferable permit fee is $200? Calculate her producer surplus with this sunk cost.
5. Suppose Megan produces T-shirts and hats printed with "I ? ECON" and she is a price taker. Megan's cost function for T-shirts is with marginal cost. Her cost function for hats is with marginal cost. The price of a T-shirt is $18 and the price of a hat is $10. What are Megan's profit-maximizing quantities of shirts and hats?
Chapter 10
Sample Problems
. Suppose you won the lottery and have the option of taking your prize in one lump sum payment of $125 million minus 8% of the lump sum, or by getting payments of $6.25 million for the next 20 years. Assuming an interest rate of 4%, which option is the better choice?
Answer:
To calculate which option is your best choice, use the formula for PDV for a constant stream:
20 year payment option:
Lump Sum:
Therefore, the best option is the 20 year payment option.
2. Suppose you are buying a car that costs $18,000. The dealer offers you a payment plan that consists of a $0 down-payment with annual payments of $3,000 for the next 6 years at a 7% interest rate. Another dealer has offered you the same car for $18,750, with a payment plan that consists of making a $3,750 down-payment with annual payments of $3,000 for the next 5 years at a 7% interest rate. Which dealer is offering the better deal?
Answer:
To figure out which dealer is offering the best deal, calculate the PDV of each offer:
For the 1st offer:
For the 2nd offer:
So, the first dealer is offering the better deal for the car.
3. Danny's baby nephew Alex is about to celebrate his 1st birthday. He decides that he wants to insure that Alex has money for college, so he invests money in a college fund that will belong to Alex on his 18th birthday. How much money would Danny have to invest today so that there is a balance of $10,000 in Alex's college fund when he turns 18 years, if the interest rate is 7%?
Answer:
To solve this problem use the formula for present discounted value:
The formula is adapted for the fact that Danny will only make a one-time payment in today's dollars, and 17 years from now Alex will have $10,000 in his college fund.
Danny will need to deposit $3.165.74 today.
4. Over the next 4 years you plan to save money in a savings account so that you can go on a backpacking trip through Europe when you graduate. You plan to save $350 freshman year, $400 your sophomore year, $450 your junior year, and $500 your senior year. Your bank gives you an interest rate of 2.5% on your savings. In today's terms, how much money will you have saved when you graduate?
Answer:
To calculate how much in present dollars you will have in your savings account when you graduate, use the formula for PDV:
In today's terms you will have saved $1,593.03.
5. Craig has the opportunity to invest in one of two possible projects. Project A requires an initial investment of $5,000 in the first year and will produce a positive net cash flow of $1,500 a year for the next 5 years. Project B requires an initial investment of $5,500 in the first year and will produce a positive net cash flow of $2,000 a year for the next 3 years. Assuming an interest rate of 8%, calculate the net present value (NPV) of each project. Calculate the payback period. Which project should Craig choose? How might your answer change if the interest rate increased to 9.5%?
Answer:
To calculate the (NPV) of each project use the formula:
For project A:
For project B:
The payback period for project A is greater than 4 years but less than 5 years.
The payback period for project B is greater than 3 years but less than 4 years.
The payback periods can be found by figuring out at what T value the investment will recoup the initial investment:
Therefore, at an interest rate of 8% project B is the better investment because of the higher NPV and a faster payback period.
If the interest rate was to increase to 9.5%, project B would still be the better investment with a higher NPV than project A, at $908.96 and $759.56 respectively.
Additional Exercises for Chapter 10
. When considering saving, borrowing and interest, what is compounding and why is it important to factor compounding into a saving or borrowing decision?
2. Calculate the present discounted value (PDV) of $1 received in one year with an interest rate of 5%, 10% and 20%. Plot your findings on the graph below. What is the relationship between the PDV and the interest rate? Why?
3. Calculate the PDV of $100 received in 5 years when the interest rate is 5%, 10%, and 20%.
4. Calculate the PDV of $1 received in one year, 5 years, 10 years, and 20 years if the interest rate is 5%.
5. Suppose you borrow $1000 today from a friend and agree to the following payment plan: $200 in one year, $250 in two years, $300 in three years, and $400 in four years. If the interest rate is 5%, how much do you owe in today's dollars?
6. You just bought a new car and have agreed to make payments of $5000 annually for 6 years starting one year from now. If the interest rate is 7%, how much do you owe in today's dollars? How does that compare to the unadjusted payment of $30,000? Explain.
7. Suppose you are offered the following options for your new car purchase. (1) Pay $25,000 now, (2) Make annual payments of $9,000 for 3 years, or (3) Make annual payments of $6,000 for 5 years. If the interest rate is 8%, which option offers the lowest price in current dollars?
8. Calculate the PDV of promised payments for a 10 year bond with a coupon of $50 and a face value of $5,000 at an interest rate of 5%. Repeat this calculation with an interest rate of 10% and 15%. How would your answers change if the coupon was reduced to $0?
9. If the nominal interest rate is 6.5 percent and the rate of inflation is 2 percent, what is the real interest rate? What would happen if the rate of inflation grew to 3.5 percent?
0. If the real interest rate is 4 percent and the inflation rate is 3.5 percent, what is the nominal interest rate?
1. Assume that Brian will earn $1000 this year and nothing next year. Considering his preferences given by his indifference curve in the graph below, will Brian choose to save or borrow to maximize his preferences? How do you know?
2. Using the income and substitution effects, explain how an increase in the interest rate could either increase or decrease a consumer's savings rate.
3. Suppose Brian's only expense is food and he plans his consumption for two year periods. His food consumption this year is F0, his food consumption next year is F1, and his marginal rate of substitution across years is. Suppose Brian earns $2,400 this year and nothing next year. If food costs $1 per pound and the interest rate is 5%, how much will Brian consume this year and how much will he save? How would your answer change if Brian now earned $2,400 this year and $1,200 next year? Write a formula for his saving as a function of the interest rate.
4. Suppose Brian's only expense is food and he plans his consumption for two year periods. His food consumption this year is F0, his food consumption next year is F1, and his marginal rate of substitution across years is. Suppose Brian earns nothing this year and $2,400 next year. If food costs $1 per pound and the interest rate is 5%, how much will Brian consume this year and how much will he save? Write a formula for his saving as a function of the interest rate.
5. Suppose Brian's only expense is food and he plans his consumption for two year periods. His food consumption this year is F0, his food consumption next year is F1, and his marginal rate of substitution across years is. Suppose Brian earns $2,000 this year and $1,000 next year. If food costs $1 per pound and the interest rate is 5%, how much will Brian consume this year and how much will he save? Write a formula for his saving as a function of the interest rate.
6. You have the opportunity to invest in one of two possible projects. Project A requires an initial investment of $15,000 in the first year and will produce a positive net cash flow of $5,000 a year for the next 4 years. Project B requires an initial investment of $10,000 in the first year and will produce a positive net cash flow of $4,350 a year for the next 3 years. Assuming an interest rate of 10%, calculate the net present value (NPV) of each project. Calculate the payback period and the internal rate of return. Which project would you choose? How might your answer change if the interest rate increased to 12%?
Chapter 11
Sample Problems
. Suppose you have a loaded coin that increases the chances of heads appearing from 50% to 75%. Unaware of the existence of your loaded coin, your younger sibling proposes a bet in which he selects tails. The bet promises $5 to the winner, to be paid by the loser of the bet. What is your expected payoff? What is your sibling's expected payoff?
Answer:
To find your expected payoff set up the following function:
Your siblings expected payoff function is:
Your expected payoff is $3.75 and your sibling's is $1.25, however, you should tell him about your unfair advantage.
2. Suppose that Penelope sells pretzels at the local professional basketball stadium. If the team wins enough games they will go to the playoffs providing Penelope withdollars in profit, if they lose Penelope will havedollars. The probability of the team winning and Penelope receivingin profit is, and the probability of losing and receiving is, making the expected profit, such that. Ifis $250, derive the expected profit line. What is the slope of this line if? What if?
Answer:
To derive the constant expected profit line, set up the expected profit equation with EP = $250, and proceed to solve forin terms ofand.
Solving for:
If:
The slope of the line is negative two-thirds.
If:
The slope of the line is negative 3/5.
3. Suppose Penelope's indifference curve is given by. , and. If the probability of a win, , is, what is Penelope's expected profit? What is the certainty equivalent of this risky bundle? What is the risk premium?
Answer:
Penelope's expected profit is. To calculate the certainty equivalent, find the level of guaranteed profit, P, that places Penelope on the same IC curve as the risky bundle.
So the certainty equivalent is $139.24. Since, the risk premium is $5.76.
4. Again, Penelope has achance of earningand achance of. His indifference curve is given by a formula. For these indifference curves, the marginal rate of substitution is. Without insurance, Penelope can profit $256 if the team wins and $0 if the team loses. She can purchase insurance for a premium of 50 cents for each dollar of promised benefit. Will Penelope buy insurance? How much will it cost and what is the value?
Answer:
Penelope's available choices lie on the budget line. In this case R = 0.5, so the slope of the budget line is -l. The formula for Penelope's budget line is thuswhere C is a constant. Penelope's budget line must pass through the point,, we know thatso C =256.
At any best choice point, the slopes of the budget line and the indifference curve must be the same. Therefore,
Using this formula, then solve forwhich gives you,.
Now we can use bothandto find Penelope's best choice, by solving algebraically.
Penelope profits $240.94 if the team wins. Since Penelope started with $256 he did buy insurance. She spent $15.06 on insurance. This means that Penelope's insurance benefit must be $30.12 (since ). If Penelope gets the low state, she profits.
After purchasing insurance, Penelope's risky bundle is. Points on the indifference curve that run through this bundle satisfy the formula. To find a risk-less bundle on this indifference curve (where), we solve:
B = $173.98 is the certainty equivalent of Penelope's risky bundle after purchasing insurance.
Penelope's initial bundle is. This bundle satisfies the formula. Like before we need to find the risk-less bundle on this indifference curve, by solving:
$163.84 represents the certainty equivalent of Penelope's initial risky bundle. The value of the insurance is $10.14, since.
5. The Super Bowl is around the corner and Andy is looking to bet on the game. He has $1000 and seeks to maximize his expected benefit,. He can either keep his $1000 or he can invest the $1000 on Team A that has a 25% chance of winning but would give him a return of $2500, and a 75% chance of losing leaving Andy without money. Andy can also bet on Team B which has a 75% chance of winning giving Andy a return of $1500. Betting on Team B has a 25% chance of returning $0 (if Team B losses). Andy has a final option which consists of betting $600 on Team B and $400 on Team A, which will earn him $900 if Team B wins and $1000 if Team A wins. Rank Andy's options from most profitable to least profitable. Which should he choose?
Answer:
If Andy keeps the $1000, he is guaranteed a benefit of 251.19 (), and a certainty equivalent of $1000. Betting $1000 on Team A will produce an expected benefit of,
Andy's certainty equivalent is the amount X that solves, so X =441.92.
If Andy bets $1000 on Team B this will produce an expected benefit of,
Andy's certainty equivalent is the amount X that solves, so X =1046.95.
If Andy bets $400 on Team A and $600 on Team B this will produce an expected benefit of:
Andy's certainty equivalent is the amount X that solves, so X =$924.80. So Andy's best option would be to bet $1000 on Team B to win. His second best option would be the hedging bet of betting on both teams. The least preferable choice would be to bet $1000 on Team A.
Additional Exercises for Chapter 11
. Assume that a coin flip is equally likely to land on heads (with probability 0.5) or tails (with probability 0.5). If the state is heads, you win $1 and if the state is tails, you lose $1. Calculate your expected payoff of one coin flip. What is your total expected payoff of flipping the coin 10 times?
2. The dreaded check engine light comes on and you take your car to a mechanic. The mechanic tells you there is a 70% chance that it could be the battery, a 20% chance it's the fuel injection, and a 10% chance that the engine will need to be replaced. Replacing the battery costs $150, fixing the fuel injection costs $600, and replacing the entire engine would cost $1,450. What is your expected cost of repairs? What if this mechanic used a subjective probability and the real probabilities are 60%, 25%, and 15%, respectively? Now what is the expected cost of repairs?
3. Suppose that Michael is a gambling man and has placed his bets accordingly. If Michael wins, he'll havedollars and if he loses, he'll havedollars. The probability of winning and havingis ? and the probability of losing and receivingis 1-?, making the expected consumption,, such that. If the expected consumption is $400, derive the constant expected consumption line,. What is the slope of this line if? What if?
4. Suppose Michael's indifference curve is given by. and. If the probability of a win,, is, what is Michael's expected consumption? What is the certainty equivalent of this risky bundle? What is the risk premium?
5. Again, Michael's indifference curve is given by. and. If the probability of a win,, is, what is Michael's expected consumption? What is the certainty equivalent of this risky bundle? What is the risk premium?
6. Suppose Michael's preferences can be represented according to the expected utility function. The probability of Michael winning his bet and earningisand the probability of Michael losing his bet and earning is. Graph Michael's indifference curve that intersects the point . Graph the constant expected consumption line through (100,100). Would Michael's risk preferences be described as risk averse, risk neutral, or risk loving?
7. Michael's preferences are now represented according to the expected utility function. The probability of Michael winning his bet and earningisand the probability of Michael losing his bet and earning is. Graph Michael's indifference curve that intersects the point . Graph the constant expected consumption line through (100,100). Would Michael's risk preferences be described as risk averse, risk neutral, or risk loving?
8. Michael's preferences are now represented according to the expected utility function. The probability of Michael winning his bet and earningisand the probability of Michael losing his bet and earning is. Graph Michael's indifference curve that intersects the point . Graph the constant expected consumption line through (100,100). Would Michael's risk preferences be described as risk averse, risk neutral, or risk loving?
9. The graph below shows the expected utility function for Michael according to his preferences, . The probability of Michael winning his bet and earningisand the probability of Michael losing his bet and earning is. Using the valuesand, first identify the benefit from each state. Next, calculate the expected consumption and identify on the graph. Finally, illustrate the certainty equivalent and the risk premium on the graph.
0. What does it mean for an insurance policy to be actuarially fair? Why would someone be willing to purchase insurance that is actuarially fair?
1. Michael has achance of winningand achance of winning. His indifference curves are given by the formula. For these indifference curves, the marginal rate of substitution is. Without insurance, Michael can spend $324 if he wins and $64 if he loses. He can purchase insurance for a premium of 40 cents for each dollar of promised benefit. Will Michael buy insurance? How much will it cost and what is the value?
2. Again, Michael has achance of winningand achance of winning. His indifference curves are given by the formula. For these indifference curves, the marginal rate of substitution is. Without insurance, Michael can spend $400 if he wins and $0 if he loses. He can purchase insurance for a premium of 40 cents for each dollar of promised benefit. Will Michael buy insurance? How much will it cost and what is the value?
3. What is the difference between hedging and diversification? Briefly explain how each can be used to mitigate risk.
4. Michael has $400 and seeks to maximize his expected benefit,. He can either keep the $400 or he can invest $200 in a company that will double his investment with probability 0.5, with a 50% chance of failing and paying out $0. Calculate the certainty equivalent of each option. Which option will Michael choose?
5. Michael has $400 and seeks to maximize his expected benefit,. He can either keep the $400 or he can invest the $400 in a company that has a 30% chance of doubling his money ($800), a 40% chance of returning even money ($400), and a 30% chance of failing and paying out $0. Which option will Michael choose? Is there a level of investment under $400 that would improve the investment option?
Chapter 12
Sample Problems
. Suppose there are two coffee shops servicing the local university, but they are own by two separate firms. Both vendors can price either high or low prices. Use the payoff matrix below to find the Nash equilibrium, if it exists. Explain how you found it. If you found a Nash equilibrium, was it the best possible outcome for both firms?
FIRM 1
FIRM 2
Answer:
To find the Nash equilibrium we start by looking at one firm's best response when the other firms choice is held constant:
Let's look at Firm 1:
If Firm 2 chooses, then Firm 1 will prefer to price low.
If Firm 2 chooses, then Firm 1 will prefer to price low.
Let's looks at Firm 2:
If Firm 2 chooses, then Firm 1 will prefer to price low.
If Firm 2 chooses, then Firm 1 will prefer to price low.
Therefore the Nash equilibrium is for both firms to price low. This outcome is not the best possible outcome for both firms. They could have done better if both would price high.
2. What is the Winner's curse and in which situations is it likely to arise?
Answer:
The winner's curse is a phenomenon in certain types of auctions, in which unsophisticated bidders tend to overpay whenever they win. The winner's curse is likely to arise whenever an item with a commonly perceived value depends on information that may not be known to all bidders.
3. Danny and Krystin are having a friendly disagreement about what to do for fun tomorrow. They agree that the best way to solve the dispute is with a game of heads and tails. Both will each place a coin down facing the side of their choosing simultaneously and if the coins match (heads-heads, or tails-tails) then Krystin wins and John pays for a night of ballroom dancing. If the coins come up different (heads-tails, or tails-heads), then Krystin pays to go to the car show. Use the table below to indicate the Danny's and Krystin's best responses. Is there a Nash equilibrium? Are any strategies strictly or weakly dominated? Solve for mixed strategies.
Krystin
Heads
Tails
Danny
Heads
Tails
Answer:
For pure strategies, there are no Nash equilibria, and also, every strategy is strictly dominated.
For mixed strategies, in equilibrium a player will not choose a dominated strategy as a best response. In the matrix above every choice can be dominated by the other player's response, therefore Danny and Krystin must randomize their choices. Let Q represent the probability that Danny selects heads. Let P represent the probability that Krystin selects heads. Danny's expected payoff from selecting heads is , while his expected payoff from selecting tails is. Since Danny's choice of P has to make Krystin indifferent between selecting heads or tails (so that she's willing to randomize between the two), we know that , which means that. Krystin's expected payoffs are similar to Danny's: For selecting heads her expected payoff isand for selecting tails her expected payoff is. Equating the two expected payoffs we get that. Therefore, the mixed strategy equilibrium is: Danny selects heads and tails with 50% probability; Krystin selects heads and tails with 50% probability.
4. Suppose the company Andy's Sandwiches is competing with Geneva's Foot-longs. Both companies must decide whether they will price their sandwiches high, medium, or low. Because Andy is well-established in the area the company makes pricing decisions before Geneva. Use backward induction on the tree below to find the Nash equilibrium.
Answer:
To find the Nash equilibrium using backward induction we must first focus on Geneva's choices holding Andy's choices constant. If Andy prices low initially, Geneva's best option is to price low (payoff = 100). If Andy prices medium initially, Geneva's best choice is to price high (payoff = 160). If Andy prices high, Geneva's best choice is to price low (payoff = 250). Now, we can focus on Andy's best decision based on what we know Geneva will do. If Andy prices low, given what we know of Geneva, his payoff will be 100. If Andy prices medium, given what we know of Geneva, his payoff will be 80. If Andy prices high, given what we know of Geneva, his payoff will be 50. As a result, the Nash equilibrium is for both Andy's Sandwiches and Geneva's Foot-longs to price low.
Additional Exercises for Chapter 12
. The matrix below describes the game of Chicken. Two players drive their car down the center of the road towards each other and each must choose whether to STAY or SWERVE. Staying wins peer approval and a payoff of 4 if the other player chooses to SWERVE. Swerving makes you look like a chicken (hence the name of the game) if the other player STAYS, and you lose popularity equal to payoff -2. However, if both players decide to STAY, then they crash and each receives -6. If they both SWERVE, then at least they tried, although each receives -2. Use this information to fill in the matrix.
PLAYER A
STAY
SWERVE
PLAYER B
STAY
SWERVE
2. The matrix below describes two firms deciding which price level to set: low price, middle price, or a high price. If they both set high prices, they each earn $500 million. If they both set middle prices, they each earn $250 million. If they both set low prices, they each earn $100 million. When the prices diverge, the firms receive payoffs as following: ,, . Assume the payoffs are symmetric, use this information to fill in the matrix.
FIRM A
FIRM B
3. The matrix below describes a game between two people attempting to maximize their payoffs. Player A has two possible options: left or right. Player B has two possible options: up or down. Does either player have a dominant strategy? Which one(s)? Explain how you know.
PLAYER A
LEFT
RIGHT
PLAYER B
UP
DOWN
4. What does it mean for a dominant strategy to be weakly dominated? How is this different from a strictly dominated strategy?
5. The matrix below describes a game between two people attempting to maximize their payoffs. Player A has two possible options: left or right. Player B has two possible options: up or down. Find the Nash equilibrium or explain why one does not exist.
PLAYER A
LEFT
RIGHT
PLAYER B
UP
DOWN
6. The matrix below describes a game between two firms attempting to maximize their profits. Each firm must choose whether to set a low price or a high price. Find the Nash equilibrium or explain why one does not exist. How is this game similar to a Prisoner's Dilemma?
FIRM A
FIRM B
7. Two students, Nina and Margaret, have been paired up to complete a problem set for their Microeconomics class. They decide to each take half of the problems and then combine their answers when they turn in the assignment. They do this simultaneously and neither will know how hard the other has worked until they receive their score. When Nina works X hours and Margaret works Y hours, they receive a benefit of . The marginal benefit of spending extra time is , regardless of who puts in the additional time. The cost to Nina is, making her marginal cost equal to. The cost to Margaret is, making her marginal cost equal to. What is the Nash equilibrium? Is there a more preferred outcome for Nina and Margaret?
8. Nina and Margaret have been paired up to complete a problem set for class. They decide to each take half of the problems and then combine their answers when they turn in the assignment. They do this simultaneously and neither will know how hard the other has worked until they receive their score. When Nina works X hours and Margaret works Y hours, they receive a benefit of . The marginal benefit of spending extra time for Nina isand the marginal benefit for Margaret is . The cost to Nina is, making her marginal cost equal to. The cost to Margaret is, making her marginal cost equal to. What is the Nash equilibrium? Is there a more preferred outcome for Nina and Margaret?
9. What is a mixed strategy equilibrium? In what type of games would this type of strategy be used?
0. Instead of dividing up the problem set, Nina and Margaret decide to play a game of odds and evens to make one person complete the entire homework assignment. In this game, each player holds up either one or two fingers at the same time as the other player. If the sum of the number of fingers is even, Nina loses and has to do the entire problem set. If the sum of the number of fingers is odd, Margaret loses and has to do the problem set. The game and payoffs in terms of an increase/decrease in utility for each person are shown in the matrix below. Find all of the Nash equilibria in pure and mixed strategies.
MARGARET
One finger
Two fingers
NINA
One finger
Two fingers
1. Again, Nina and Margaret decide to play a game of odds and evens to make one person complete the entire homework assignment. In this game, each player holds up either one or two fingers at the same time as the other player. If the sum of the number of fingers is even, Nina loses and has to do the entire problem set. If the sum of the number of fingers is odd, Margaret loses and has to do the problem set. Since Nina would spend more time on the problem set, it is a worse outcome for her if she loses, although it is a better outcome for Margaret since they get a higher score. The game and payoffs in terms of an increase/decrease in utility for each person are shown in the matrix below. Find all of the Nash equilibria in pure and mixed strategies.
MARGARET
One finger
Two fingers
NINA
One finger
Two fingers
2. The tree below describes a multi-stage game of Chicken, assuming that Player A can see what Player B is going to do just before they must decide. The two players drive their cars down the center of the road towards each other and each must choose whether to STAY or SWERVE. Staying wins peer approval and a payoff of 4 if the other player chooses to SWERVE. Swerving makes you look like a chicken (hence the name of the game) if the other player STAYS, and you lose popularity equal to payoff -2. However, if both players decide to STAY, then they crash and each receives -6. If they both SWERVE, then at least they tried, although each receives -2. Solve for the Nash equilibrium using backward induction.
3. Centipede is a well-known game where two players have a chance to "grab a dollar." Player 1 moves first and Player 2 moves second. The game begins with $1 sitting on a table. Player 1 can either take the $1 or wait. If Player 1 takes the $1 the game is over, and Player 1 gets to keep the dollar. If Player 1 waits, the $1 quadruples to $4. Now it is Player 2's turn. Player 2 can either take the entire $4 or split is evenly with Player 1, $2 each. If Player 2 waits in the second round, then the money on the table quadruples again and Player 1 can now take it all or split it evenly, $8 each. Draw the game tree and solve for the equilibrium using backward induction. Would allowing additional rounds change the equilibrium outcome? Why or why not?
4. What is a grim strategy in multiple stage games? Could a grim strategy be used in the game of Chicken described in question #12? Why or why not?
5. The matrix below describes a game between two firms attempting to maximize their profits. Each firm must choose whether to set a low price or a high price. Given the rules and payoffs, first solve for the Nash equilibrium. Could a grim strategy be devised to help foster cooperation between the two firms? What is the goal and how could it be accomplished?
FIRM A
FIRM B
Chapter 13
Sample Problems
. What is Neuroeconomics? How could the study of the neural process help with economic modeling?
Answer:
Neuroeconomics is a new field that that studies the human neural system, with the purpose of discovering new principles of economic decision making. Neuroeconomics research studying the issue of self-control has identified a likely source of dynamic inconsistency. Dynamic inconsistency is something that standard economic theory has not accounted for traditionally, so knowing the source of this phenomenon would undoubtedly help make economic models involving a consumer's choices (for example) more realistic.
2. What is the trust game and why is it important to research in the field?
Answer:
A trust game involves two players, in which one player (the trustor) who starts out with a fixed amount of money, which he can keep or invest. The second player (the trustee) divides up the principal and earnings. The trust game has real-world applications to situations where legally binding contracts don't work well. The trust game helps economists understand one reason why this approach works.
3. What is the voluntary contribution game and what is the possible shortcoming of game theory that it highlights?
Answer:
The voluntary contribution game is a multi-player game in which each member has a choice whether or not to contribute to a common pool. Each player's contribution benefits everyone, but the contributor's cost exceeds the player's individual return. Results from two-stage voluntary contribution games have shown that despite the game theoretic best response being zero contribution, the contribution rate in experiments continued to increase throughout rounds. Game theory assumes that players care only about their own monetary rewards, but in fact players may also care about other issues, like fairness.
4. What is the sunk cost fallacy and how might this apply to why people buy expensive exercise equipment that they won't use but are unwilling to throw it away?
Answer:
The sunk cost fallacy refers to the belief that, if you paid more for something, it must be more valuable to you. When people are reluctant to throw away expensive exercise equipment, this is an example of how people fall for the sunk cost fallacy. Some exercise equipment is so expensive that people find it hard to throw it away because the high price paid to acquire it is tied to the worth of the object, despite the fact that they will probably never use it.
5. Adriana evaluates her risky options based on prospect theory according to the following equation: whereandare the potential payoffs associated with the probabilitiesand, respectively. For positive values of X, her valuation function is. For negative values of X, her valuation function is. Adriana's probability weighting function is. Would Adriana take an investment that pays $1000 with probability and -$1000 with probability?
Answer:
We need to solve for the expected valuation of each risky prospect using expressions for positive and negative values of X. If X = 1000, then . The weight attached tois, while the weight attached to is .The value of this option is.
Adriana will take part in the investment in question because it offers a positive payoff.
Additional Exercises for Chapter 13
. Briefly explain how experiments can be used to gain additional insight into human behavior beyond the standard economic theories.
2. What are two advantages and two disadvantages with economic experiments? What are two methods used to overcome the disadvantages of laboratory experiments?
3. Suppose that you are offered the following options: $100 with certainty or a gamble with a 1 in 10 chance of winning $900 and a 9 in 10 chance of winning $0. Being a college student, you decide to go with the guaranteed cash and take the $100. However, when asked to state a dollar amount that is just as good as the gamble, you answer $150. How does this violate standard economic theory? Why might this still be considered rational behavior?
4. Briefly explain the concept of anchoring. How might choosing lottery numbers based on birthdates, anniversaries, etc. fall under this category?
5. Suppose your professor has offered three options for the final exam: a take-home exam, an in-class exam, or a term paper. The professor wants to allow students to choose the option that maximizes their skills, but the professor would also prefer for most students choose the in-class exam because it is the easiest to grade. How could this professor use their knowledge of the default effect to direct students towards the in-class exam? How do you know this would likely be effective?
6. What is the endowment effect and how does it demonstrate a bias towards the status quo?
7. Suppose that Jason is shopping for a new television and DVD player. The store closest to Jason has the television for $3,500 and the DVD player for $300. Now consider two different scenarios. In the first scenario, the salesman tells Jason that the television is on sale for $3,350 at a store 10 miles away. In the second scenario, the salesman tells Jason that the DVD player is on sale for $150 at a store 10 miles away. Using the concept of narrow framing, explain why Jason would be more likely to take advantage of the sale in the second scenario and why.
8. What is one possible solution for people who are dynamically inconsistent? Apply this solution to the case of someone who is trying to quit smoking.
9. Every week, Brian has to give his cat Vincent a bath. It's a frustrating task and on the day he does it, his utility is reduced by 6. However, having Vincent nice and clean increases Brian's utility on the following day. If he bathes Vincent on Monday, his payoff is 16; if he does it Tuesday, his payoff is 13; if he does it Wednesday, his payoff is 10; if he does it Thursday, his payoff is 7; and if he does it Friday, his payoff is 4. Each day he attempts to maximize his utility according to the utility function, where is his payoff on that day andis the sum of his payoff on future days. As of Sunday, what day would Brian pick to wash Vincent? Will he actually choose to wash Vincent on that day? Why or why not?
0. Every week, Brian has to scoop his cat's litter box. It's an unpleasant task and on the day he does it, his utility is reduced by 2. However, a clean litter box is much nicer than a dirty one and this increases Brian's utility on the following day. If he scoops the box on Monday, his payoff is 10; if he does it Tuesday, his payoff is 8; if he does it Wednesday, his payoff is 6; if he does it Thursday, his payoff is 4; and if he does it Friday, his payoff is 2. Each day he attempts to maximize his utility according to the utility function, where is his payoff on that day andis the sum of his payoff on future days. As of Sunday, what day would Brian pick to scoop the litter box? Will he actually choose to scoop it on that day? Why or why not?
1. Again, every week Brian has to scoop his cat's litter box. It's an unpleasant task and on the day he does it, his utility is reduced by 2. However, a clean litter box is much nicer than a dirty one and this increases Brian's utility on the following day. If he scoops the box on Monday, his payoff is 10; if he does it Tuesday, his payoff is 8; if he does it Wednesday, his payoff is 6; if he does it Thursday, his payoff is 4; and if he does it Friday, his payoff is 2. Each day he attempts to maximize his utility according to the utility function, where is his payoff on that day andis the sum of his payoff on future days. As of Sunday, what day would Brian pick to scoop the litter box? Will he actually choose to scoop it on that day? Why or why not?
2. Use the concept of a projection bias to explain why people tend to wish they'd had even just 5 more minutes with a loved one after they pass away.
3. Suppose Jason takes a trip to Las Vegas and begins his weekend playing roulette. In this game, he bets either red or black. Assume that red and black are equally likely to hit, so the ball will land on a red number 50% of the time and a black number 50% of the time. Jason comes across a table that has hit 10 red numbers in a row. If he believes in a gambler's fallacy, will he bet on red or black next? What will he bet if he believes in the hot-handed fallacy? Is one bet better than the other?
4. Brian evaluates his risky options based on prospect theory according to the following equation: whereandare the potential payoffs associated with the probabilitiesand, respectively. For positive values of X, his valuation function isand for negative values of X, his valuation function is. Brian's probability weighting function is. Would Brian take part in a gamble that pays $10 with probability and -$10 probability? Would he accept a gamble that pays $100 and -$100 with equal probability?
5. Brian evaluates his risky options based on prospect theory according to the following equation: whereandare the potential payoffs associated with the probabilitiesand, respectively. For positive values of X, his valuation function isand for negative values of X, his valuation function is. Brian's probability weighting function is. Would Brian take part in a gamble that pays $10 with probability and -$10 probability? Would he accept a gamble that pays $100 and -$100 with equal probability?
6. What is the difference between the dictator game and the ultimatum game? How do these games allow for emotional behavior?
Chapter 14
Sample Problems
. Daphne's demand for breakfast burritos is below $6, and zero at prices above $6. Joaquin's demand function isat prices below $4, and zero at prices above $4. What is the market demand function?
Answer:
At prices below the $4 the market demand is . For prices between $4 and $6, only Daphne purchases breakfast burritos, so the market demand function is. At prices above $6, neither Joaquin nor Daphne will buy burritos.
2. What are the three properties of long-run competitive equilibrium with free entry, and why is it that a firm earning zero economic profit is not necessarily doing poorly?
Answer:
The three properties of long-run competitive equilibrium in a market with free entry are: the equilibrium price must be equal to the , firms earn zero profit, and each active firm produce at its efficient scale of production.
When we take economic costs into account we are including all opportunity costs into account. So, when a firm is making zero economic profit the owner(s) of the firm are being fully compensated for their all of their costs (explicit and implicit), which means that the firm cannot make more money doing anything else.
3. The market demand function for desk lamps isand the market supply function is, with both quantities are measured in millions of lamps per year. What are the aggregate surplus, consumer surplus, and producer surplus at the competitive market equilibrium? What would be the deadweight loss if the desk lamp suppliers only produced 4.5 million lamps?
Answer:
First, we must find the market equilibrium. This is done by rearranging both supply and demand function in terms of Q rather than P. Then it is just a matter of equating the two functions and solving for Q. (and).
If Q = 5.14, then.
Now, to find the consumer surplus take the area above the horizontal line at P = $7.72 but below,
To find the producer's surplus take the area below the horizontal line at P = $7.72 but above,
Aggregate surplus equals the sum of consumer surplus and producer surplus. Thus,
If the producers in this market were to produce only 4.5 million lamps, then the deadweight loss would be the area between the quantities 4.5 and 5.14 and above but below. Also we need to find P for both functions when Q = 4.5 (and ). Thus,
$0.1216 million is the deadweight loss if the producer's in this market only produce 4.5 million desk lamps per year.
4. Daphne and Joaquin both love burritos. Daphne's demand for burritos is and Joaquin's demand for burritos is. What is Daphne's willingness to pay for 6 burritos? What is Joaquin's willingness to pay for 10 burritos?
Answer:
To find Daphne's willingness to pay for 6 burritos, find the vertical intercept and evaluate at Q = 6. If Q = 6, then. The P-intercept is 18. Her willingness to pay is the area under the line from the vertical-intercept to where P = 4.
If Q = 10, then. Joaquin's willingness to pay for 6 burritos is:
Additional Exercises for Chapter 14
. What are the three main characteristics of a perfectly competitive market? How does the presence of these three factors lead to price taking behavior by firms?
2. Marie and Jerry both enjoy pizza. Marie's demand for pizza slices is and Jerry's demand for pizza slices is. Assuming they are the only people in this market, what is the market demand function? Graph the market demand curve below.
3. Marie and Jerry both enjoy pizza. Marie's demand for pizza slices is and Jerry's demand for pizza slices is. Assuming they are the only people in this market, what is the market demand function? Graph the market demand curve below.
4. The supply function for a perfectly competitive firm making slices of pizza is. Assuming there are three firms in this market, what is the market supply curve? Graph it below.
5. Suppose that a pizza firm's long run variable costs are. The marginal cost is thus. Assume the firm faces an avoidable fixed cost of $45 per day and there is free entry into the market in the long run. What is the long-run market supply curve?
6. Suppose that a pizza firm's long run variable costs are. The marginal cost is thus. Assume the firm faces an avoidable fixed cost of $25 per day and there is free entry into the market in the long run. What is the long-run market supply curve?
7. Assume the daily market demand for pizza slices is , where P is the price for one slice of pizza. The daily costs for a firm making pizza slices are variable costand an avoidable fixed cost of $45. The firm's marginal cost is. Suppose that there is free entry in the long run. What are the long-run market equilibrium price and quantity? How many firms are in this market and how many slices of pizza does each firm produce?
8. Assume the daily market demand for pizza slices is , where P is the price for one slice of pizza. The daily costs for a firm making pizza slices are variable costand an avoidable fixed cost of $25. The firm's marginal cost is. Suppose that there is free entry in the long run. What are the long-run market equilibrium price and quantity? How many firms are in this market and how many slices of pizza does each firm produce? If the market demand increases to, what is the new short-run equilibrium?
9. Suppose there is a competitive market with free entry. If demand doubles will the long-run price always remain the same? Why or why not?
0. Marie and Jerry both enjoy pizza. Marie's demand for pizza slices is and Jerry's demand for pizza slices is. What is Marie's willingness to pay for 4 slices? What is Jerry's willingness to pay for 4 slices?
1. The supply function for a perfectly competitive firm making slices of pizza is. Calculate the firm's avoidable cost of making 180 slices. What is the avoidable cost of making 220 slices?
2. What is deadweight loss and when does it arise?
3. The market demand function for down blankets isand the market supply function is. Calculate the consumer surplus, producer surplus and aggregate surplus at the competitive market equilibrium.
4. The market demand function for pizza slices isand the market supply function is. Calculate the consumer surplus, producer surplus and aggregate surplus at the competitive market equilibrium.
5. The market demand function for refrigerator magnets isand the market supply function is. Calculate the consumer surplus, producer surplus and aggregate surplus at the competitive market equilibrium.
Chapter 15
Sample Problems
. In a perfectly competitive market, how is the burden of a tax shared and what determines the incidence of tax?
Answer:
In a perfectly competitive market, the burden of a tax is shared by consumers and firms. The incidence of the tax is dependent on the elasticities of both the demand and supply curves.
2. The market demand function for beach cruiser bicycles is and the market supply function is . Suppose that the government imposes a $12 tax on each cruiser sold. What will be the effects on aggregate surplus, consumer surplus, and producer surplus? What is the deadweight loss caused by the tax?
Answer:
First find the equilibrium price without taxes, by equating supply and demand.
Consumer surplus is:140-90)(150-0)=3750. Producer surplus is:. Aggregate surplus is thus equal to $9375.
Now to find the new equilibrium with taxes, we need to find the new market supply function after taxes.
where Pb is the price paid by consumers. As before, the equilibrium level of Pb is found by equating supply and demand:
This means that sellers receive PS = $82.8 (or 94.8-12). So buyers pay $4.8 more and sellers receive $7.20. Substituting $94.8 into the market demand function tells us that 135.6 bicycles are bought and sold with the tax.
The consumer surplus after the tax is:
The producer surplus after the tax is:
Thus, aggregate surplus is 7661.4.
The deadweight loss caused by the tax is calculated as follows:
3. Suppose the government is trying to balance its budget, but needs to raise taxes so that they can cover the amount they have allocated on infrastructure expenditures. The government has decided to tax both gasoline and diary milk, but legislators are uncertain which to tax more heavily. The government is also concerned about minimizing the deadweight loss to producers and consumers. If the elasticity of demand for gasoline is -4.5, and the elasticity of demand for diary milk is -3.2 which product should the government tax more heavily?
Answer:
If the government in question was trying to minimize the deadweight loss of taxation, it would want to tax goods that have more inelastic demands and supplies. In this case, the demand for gasoline is much more inelastic than the demand for diary milk, so it follows that this government should tax gasoline much more heavily than diary milk.
4. In what situation would it be beneficial, in terms of aggregate surplus, to impose a trade barrier on an importing supplier? Explain why.
Answer:
A trade barrier can sometimes increase a domestic country's aggregate surplus if the supply curve of the importing supplier is upward sloping. For instance, if this was the case, then imposing a tariff on an importing supplier would have the same effect as taxing a domestic producer. The difference in this situation is that the domestic government will not concern itself with the loss of surplus of a foreign producer. This can sometimes produce a domestic benefit that exceeds the deadweight loss created by a trade barrier.
Additional Exercises for Chapter 15
. For each of the following, state whether the tax is a specific tax or an ad valorem tax: (1) an 18 cent tax on each gallon of gas, (2) a 7.56 percent payroll tax, (3) an 8.25 percent sales tax, and (4) a $1.25 tax on cigarettes.
2. For each of the following sets of elasticities state which group, either consumers or producers, will bear the greater burden of a small tax: (1) [5, 1.2], (2) [2, 0.2], (3) [0.8, 1], (4) [1, 1.4], (5) [1.3, 1.3].
3. The market demand function for down blankets isand the market supply function is. Suppose the government imposes a $15 tax on the sellers of each down blanket. Calculate the consumer surplus, producer surplus and aggregate surplus at the competitive market equilibrium for before and after the tax. What is the deadweight loss of the tax?
4. The market demand function for pizza slices isand the market supply function is. Suppose the government imposes a $0.30 tax on the sellers of each pizza slice. Calculate the consumer surplus, producer surplus and aggregate surplus at the competitive market equilibrium for before and after the tax. What is the deadweight loss of the tax?
5. The market demand function for refrigerator magnets isand the market supply function is. Suppose the government imposes a $0.50 tax on the sellers of each refrigerator magnet. Calculate the consumer surplus, producer surplus and aggregate surplus at the competitive market equilibrium for before and after the tax. What is the deadweight loss of the tax?
6. Which of the following sets of elasticities would result in the largest deadweight loss: (1) [5, 1.2], (2) [2, 0.2], (3) [0.8, 1], (4) [1, 1.4]? Briefly explain why.
7. The market demand function for pizza slices isand the market supply function is. Suppose the government grants an 18 cent subsidy for the sellers of each pizza slice. Calculate the consumer surplus, producer surplus and aggregate surplus at the competitive market equilibrium for before and after the subsidy. What is the deadweight loss of the subsidy?
8. The market demand function for pizza slices isand the market supply function is. Suppose the government wants to raise the price of a slice of pizza to $3.00 and is debating between a price floor, a price support program, a quota, and a voluntary production reduction program. They hire an economist (you!) to describe what each action would entail. Briefly explain how each program would work for this market and calculate the resulting deadweight loss.
9. The market demand function for pizza slices isand the market supply function is. Suppose the government wants to raise the price of a down blanket to $120 and is debating between a price floor, a price support program, a quota, and a voluntary production reduction program. They hire an economist to describe what each action would entail. Briefly explain how each program would work for this market and calculate the resulting deadweight loss.
0. The market demand function for refrigerator magnets isand the market supply function is. Suppose the government wants to raise the price of a refrigerator magnet to $2.00 and is debating between a price floor, a price support program, a quota, and a voluntary production reduction program. They hire an economist to describe what each action would entail. Briefly explain how each program would work for this market and calculate the resulting deadweight loss.
1. The market demand function for down blankets isand the market supply function is. Suppose the government wants to lower the price of a down blanket to $85. Explain how a price ceiling would achieve this goal. Graph the market and price ceiling below and calculate the resulting deadweight loss.
2. Suppose the United States is contemplating restricting the amount of sugar cane imported into the country. Explain how they could do this using a tariff or a quota. How would each action affect domestic surplus?
3. Assume the market demand function for a good isand the domestic supply function is. Suppose the good is also imported and the import supply curve is infinitely elastic at a price of $1.00. How would an import tariff of 50 cents impact social welfare?
4. Assume the market demand function for a good isand the domestic supply function is. Suppose the good is also imported and the import supply curve is infinitely elastic at a price of $75. How would an import tariff of $15 impact social welfare?
5. Assume the market demand function for a good isand the domestic supply function is. Suppose the good is also imported and the import supply curve is infinitely elastic at a price of $1.00. How would an import tariff of 25 cents impact social welfare?
Chapter 16
Sample Problems
. What is a lump sum transfer and how are they different from other taxes and transfers?
Answer:
With a lump-sum transfer, a consumer receives or surrenders a fixed amount of resources; it does not depend on the consumer's choices. Lump-sum transfers differ from other taxes and transfers in that they don't compromise efficiency because they don't distort choices.
2. Suppose the following formulas describe the supply and demand for coffee (where is in millions of pounds demanded andstands for millions of pounds supplied), PC stands for the price of coffee, and PD stands for the price of doughnuts.
Also suppose the following formulas describe the supply and demand for doughnuts (where is in millions of doughnuts demanded and stands for millions of doughnuts supplied), PC stands for the price of 16oz coffee, and PD stands for the price of a dozen doughnuts.
Solve algebraically for the general equilibrium effects. Also solve for the general equilibrium effect if there is a $1.50 sales tax on coffee.
Answer:
First we must find the market clearing formulas for both coffee and doughnuts. This is done by equating supply and demand in each market.
Coffee:
Doughnuts:
Next, we must solve for the prices that satisfy both market-clearing formulas at the same time. This is done by substituting one into the other, like so:
The price of doughnuts is therefore,
Plugging these values into either the supply or demand function, we find that in the general equilibrium, consumers buy 22.1 million cups of 16oz coffee, and 13.7 million boxes of a dozen doughnuts.
If the government imposes a $1.50 tax per cup of 16oz coffee, this changes the supply function for coffee to:
Once again equate demand and supply in the coffee market to find:
We can now proceed as before, substituting PH into PP:
The price for doughnuts is thus:
Plugging these values into either the supply or demand function, we find that in the general equilibrium, consumers buy 11.8 million cups of 16oz coffee, and 14.3 million boxes of a dozen doughnuts.
3. What is output efficiency? What does the output efficiency condition say and when does it apply?
Answer:
Output efficiency means that for allocations that satisfy exchange and input efficiency, it is not possible to make one consumer better off without harming anyone else by shifting production from one good to another.
The output efficiency condition states that an allocation satisfies the condition if, for every set of goods, every consumer's marginal rate of substitution equals the marginal rate of transformation.
4. What does exchange economy require to be efficient? Why are some allocations efficient while others are inefficient?
Answer:
In an exchange economy if an allocation is inefficient, there will surely be a potential for gains from trade. Thus, efficiency in an exchange economy results in there being no potential for gains from trade (meaning the economy is a Pareto optimum).
Allocations vary between efficient and inefficient because if all possible allocations were efficient there would be no need to trade anything. This is of course is not realistic.
Additional Exercises for Chapter 16
. What is the difference between partial equilibrium analysis and general equilibrium analysis? What is the advantage to using general equilibrium analysis?
2. What would completely general analysis require and why is this impractical?
3. Given the market-clearing curve for cake and ice cream in the graph below, what is the relationship between ice cream and cake? Explain.
4. Suppose the demand and supply functions for slices of pizza are and (where is millions of slices demanded, is millions of slices supplied, andis the price per slice of pizza). Also suppose the demand and supply functions for hamburgers are and. First find the prices and quantities of pizza slices and hamburgers sold in the general equilibrium. What happens if the government imposes a 35 cent tax per slice of pizza? Solve for the new general equilibrium prices and quantities of pizza and hamburgers.
5. Suppose the demand and supply functions for deli turkey meat are and (where is millions of pounds demanded, is millions of pounds supplied, andis the price per pound). Also suppose the demand and supply functions for deli roast beef are and. First find the prices and quantities of turkey and roast beef sold in the general equilibrium. What happens if the government imposes a 48 cent tax per pond of turkey? Solve for the new general equilibrium prices and quantities of turkey and roast beef.
6. Briefly explain what it means for an economy to be Pareto efficient. Can Pareto efficiency exist in an economy where one person owns 100% of all resources?
7. Which points on the graph below are Pareto efficient? If a point is not Pareto efficient, briefly explain why.
8. What is the difference between a process-oriented notion if equity and an outcome-oriented notion of equity? Which one is associated with a market-based system?
9. What is the difference between utilitarianism and Rawlsianism? What is difficult about implementing either outcome-oriented principle?
0. What is the first welfare theorem? How can an Edgeworth box be used to demonstrate this theorem?
1. What does the contract curve identify in an Edgeworth box? Why is this useful?
2. Marie and Jerry must divide 10 pounds of food and 10 gallons of water. Suppose Marie's preferences for food (F) and water (W) are represented by the utility function . Jerry's preferences are represented by the utility function , with a corresponding marginal rate of substitution for food with water . If the initial endowments for Marie and Jerry are , to which allocation will Marie and Jerry trade? What is the ratio of the price of food to the price of water in a competitive equilibrium?
3. Marie and Jerry must divide 10 pounds of food and 10 gallons of water. Suppose Marie's preferences for food (F) and water (W) are represented by the utility function . Jerry's preferences are represented by the utility function , with a corresponding marginal rate of substitution for food with water . If the initial endowments for Marie and Jerry are , to which allocation will Marie and Jerry trade? What is the ratio of the price of food to the price of water in a competitive equilibrium?
4. How is input efficiency similar to Pareto efficiency? What condition must be satisfied to have input efficiency?
5. What is the laissez-faire approach to efficiency and how might this fail?
6. What is the second welfare theorem and why does it reflect an optimistic point of view?
Chapter 17
Sample Problems
. Tokushige Software is a competition free company that sells computer operating system software. Suppose that its annual variable costs of and marginal costs of, where Q is the number of computer operating system software they sell per month. The quantity demand function for its operating software is. In addition, it has and avoidable cost of $25,000 per month. What is its profit-maximizing sales quantity and price?
Answer:
First, start by finding the inverse demand function. In this case inverse demand is, and its marginal revenue is found by using the formula:
We can proceed to find the its most profitable positive sales quantity and the associated price by equating MR = MC (by the quantity rule).
Substituting Q, into the inverse demand function we find that the price per unit of operating system software is
Now all that is left is to apply the shut-down rule. In doing so, observe that Tokushige Software's profit at a the price $203.84 is $167,288.46, which equals its total revenue of $391,984.32, minus the variable cost of selling 1923 units of software and the $25,000 avoidable cost. Since Tokushige Software earns nothing by shutting down, it will prefer to stay in business at its profit-maximizing price of $203.84.
2. What is the deadweight loss from monopoly pricing in sample problem (1)?
Answer:
To calculate the deadweight loss we need to first calculate the competitive equilibrium price and quantity. This can be found by setting equating marginal cost and the demand function for computer operating system software.
=
Thus, price in the competitive equilibrium is:
From question (1) we retrieve that the monopolistic profit-maximizing equilibrium is Q = 1923 and P = $203.84. We are now set to find the deadweight loss (Note: this is a simple matter in this example because the MC curve and demand curve are linear functions).
3. What is the pass-through rate and in what units is it measured? Suppose the demand curve has a constant elasticity of, what can you say about the price a monopolist would charge in this market?
Answer:
The pass-through rate represents an increase in the price that occurs due to a small increase in the marginal cost, measured per dollar of increase in marginal cost. A monopolist firm would set a price of 1.5 times the marginal cost, because the monopolist's price is a multiple of its marginal cost.
4. Why would it be profitable for a firm to use advertising? Would advertising be useful to a firm in a competitive market? What in controversial about the welfare analysis of advertising?
Answer:
Under certain circumstances advertising would help a firm differentiate their product or perhaps inform consumers about their product and its benefits. In this case advertising might be useful. However, it would not be useful for a firm in a competitive market because each firm perceives itself as capable of selling as much as it wants at the market price.
The welfare analysis of advertising is divisive in that the evaluation of its effect on aggregate surplus depends on why advertising has succeeded in shifting demand. If advertising is informing people about a product, then the shift in demand will not affect aggregate surplus negatively. If on the other hand, advertising is being done to deceive and mislead consumers, this will cause a negative effect on social welfare despite the shift in demand.
Additional Exercises for Chapter 17
. What characteristic identifies a firm as possessing market power? How does this apply to a monopolist?
2. The graph below shows the demand function, marginal revenue, and marginal cost for a particular market. What quantity will a profit-maximizing monopolist choose to produce? What price will they charge? What price would result in a competitive equilibrium?
3. Boiler Pizza is a local monopolist selling pizza slices. They face the hourly demand function. What is Boiler Pizza's marginal revenue when it sells 5 slices? What price would Boiler Pizza set when selling 5 slices? Graph the demand and marginal revenue curves below.
4. DownTown blankets is a local monopolist selling handmade winter blankets. They face the demand function. What is DownTown's marginal revenue when it sells 100 blankets? What price would DownTown set when selling 100 blankets? Graph the demand and marginal revenue curves below.
5. Boiler Pizza is a local monopolist selling pizza slices. They face the hourly demand function. Suppose that Boiler Pizza has variable costs and marginal cost. What is its profit-maximizing sales quantity and price?
6. DownTown blankets is a local monopolist selling handmade winter blankets. They face the demand function. Suppose that DownTown has variable costs and marginal cost. DownTown also has an avoidable fixed cost of $5,000. What is its profit-maximizing sales quantity and price?
7. Assume a monopolist decides to sell 500 units at a price of $10.00 and has a marginal cost . Use the Lerner Index to calculate the mark-up and elasticity of demand when.
8. Assume a monopolist decides to sell 8,500 units at a price of $55 and has a marginal cost . Use the Lerner Index to calculate the mark-up and elasticity of demand when.
9. Boiler Pizza is a local monopolist selling pizza slices. They face the hourly demand function. Suppose that Boiler Pizza has variable costs and marginal cost. Given its profit-maximizing sales quantity and price found in #5, calculate the deadweight loss.
0. DownTown blankets is a local monopolist selling handmade winter blankets. They face the demand function. Suppose that DownTown has variable costs and marginal cost. DownTown also has an avoidable fixed cost of $5,000. Given its profit-maximizing sales quantity and price found in #6, calculate the deadweight loss.
1. What is rent-seeking and how could it result in additional losses beyond the standard deadweight loss from a monopoly market?
2. Marty Mart is a monopsonist importer of household products specializing in toaster ovens. The market supply of toaster ovens is. What is Marty Mart's marginal expenditure when it buys 2000 toaster ovens? What is the price per toaster oven when?
3. Now assume the market supply of toaster ovens for the monopsonist Marty Mart is. What is Marty Mart's marginal expenditure when it buys 2000 toaster ovens? What is the price per toaster oven when?
4. Marty Mart is a monopsonist importer of household products specializing in toaster ovens. The market supply of toaster ovens is. Suppose that the demand for its toaster ovens is. What is the profit-maximizing price and quantity of toaster ovens? What price and quantity would result in a competitive equilibrium?
5. Now assume the market supply of toaster ovens for monopsonist Marty Mart is. Suppose that the demand for its toaster ovens is. What is the profit-maximizing price and quantity of toaster ovens? What price and quantity would result in a competitive equilibrium?
6. What is a natural monopoly? How would the government regulate a natural monopoly using first-best or second-best regulation?
Chapter 18
Sample Problems
. In terms of production and consumption, what does perfect price discrimination have in common with a perfectly competitive market?
Answer:
A monopolist produces exactly the same quantity and each consumer consumes exactly the same quantity as would occur in a perfectly competitive market. Thus with perfect price discrimination there is no deadweight loss, and markets are efficient.
2. The local tavern in a rural area acts like a monopolist. The tavern serves 250 consumers, each of whom has a weekly demand for pints of beer of , where P is the price per pint in dollars. The marginal cost of providing the beer is $1.75 per pint. If the tavern charges $3.50 per pint, how large of a fixed fee can it charge and still have customers coming back for more? What will they charge each customer? What are the monopolist's fixed fee and profit if it charges $1.75 per minute? Which of these two-part tariffs produces a larger profit?
Answer:
At a price of $3.50 per pint, each customer will buypints per week. Before the fixed fee, consumer surplus is. So the tavern could charge a fixed fee of $3.38 without discouraging consumers from buying more pints. In addition to the fixed fee profit, they make $1.75 on each of the 4.5 pints it sells to a customer, for an additional $7.88 in profit. The tavern thus makes $7.88 + $3.38 = $11.26 on each customer. Total profit is.
We need to do the same calculations for P = $1.75. pints per week. Consumer surplus before the fixed fee is. So the tavern could charge a fixed fee of $15.84 without discouraging customers from buying. Notice that the tavern will not make any profit on the pints they sell because they are pricing at their marginal cost. Total profits are. Pricing at marginal cost yields a profit $1,145 higher.
3. Suppose the monopolist tavern from the previous problem sells pints of beer to regulars (R) and to social drinkers (S). The demand function for regulars is and the demand for social drinkers is. If marginal cost per pint is $1.75 what price can the tavern set when it can and can't price discriminate? How will price discrimination affect the monopoly profit?
Answer:
First, let's find the profit maximizing price and profit associated with this price. The inverse demand function for regulars is. Recall now, what how we have been calculating marginal revenue in previous chapters.
Now set MR = MC:
Substitute the profit-maximizing quantity into the inverse demand function to find the profit-maximizing price.
Therefore profit is,.
Similarly we can find the profit-maximizing price for social drinkers. First, take the social drinkers inverse demand function and use it to find their marginal revenue function. Now set MR = MC.
Substitute the profit-maximizing quantity into the inverse demand function to find the profit-maximizing price.
Therefore profit is. Total profit for the tavern with price discrimination is $26.01.
Now we need to find the tavern's profit without price discrimination. Find the market demand, inverse market demand and marginal revenue. Market demand is calculated just as it was in previous chapters.
Inverse demand:
Marginal demand:
To find the profit-maximizing price, set MR = MC to determine which plan offers the highest profit. There are two cases to consider:
If
This quantity yields the price $3.88 and a profit of $18.06.
If
This quantity yields the price $3.66 and a profit of . Thus the profitable price is $3.66. From the demand function we know that regulars will buypints. Social drinkers will buypints.
Total profit with price discrimination is $26.01, without price discrimination $25.55 which is obviously less profitable.
4. What is mixed bundling and how could firms use mixed bundling to avoid discontinuing a low demand product?
Answer:
When a firm sells several products together as a package while also offering the same products for sale individually, this practice is known as mixed bundling. For example, inkjet cartridges are often sold as a bundle, selling both black and colored ink in one package as well as separately.
When a product is not selling well a monopolist can choose to bundle the under performing or low demand product with another that is more popular, in addition to selling the under performing product individually at a lower price. The reduced price will be less that what it would cost a monopolist to stop producing the under performing product all together. By doing this the monopolist will cater to consumers who find the bundle attractive, as well as those consumers who demand the product sold separately. Thus mixed bundling can be used to avoid discontinuing the firms under performing product.
Additional Exercises for Chapter 18
. What is perfect price discrimination? Why might this be difficult for a firm to achieve?
2. What is a two-part tariff? How is a two-part tariff an example of a type of quantity-dependent pricing?
3. If a firm can use price discrimination based on observable customer characteristics, which group will pay a higher price: consumers with low elasticity of demand or consumers with a high elasticity of demand? Explain.
4. Dolly's Pet Shop has a dog food program where pet owners can buy a special tub and then fill the tub at a reduced price per pound. There are 200 consumers, each with demand. Dolly's marginal cost per pound is $1.50. Which two-part tariff of a price per unit and initial tub fee will maximize Dolly's profit? How much will Dolly earn?
5. Dolly's Pet Shop has a dog food program where pet owners can buy a special tub and then fill the tub at a reduced price per pound. There are 200 consumers, each with demand. Dolly's marginal cost per pound is $0.50. Which two-part tariff of a price per unit and initial tub fee will maximize Dolly's profit? How much will Dolly earn?
6. Dolly's Pet Shop has two types of customers for cat food; people with a single cat (S) and those with multiple cats (M). The single-cat owners' demand for cat food is, where Q is pounds of cat food. The owners with multiple cats have demand for cat food equal to. The marginal cost is $2.00 per pound. What price per pound should Dolly set if she cannot discriminate between the two groups? If Dolly can separate the groups, what price per pound should she charge to the members of each group? Compare Dolly's profits with and without discrimination.
7. Dolly's Pet Shop still has two types of customers for cat food; people with a single cat (S) and those with multiple cats (M). The single-cat owners' demand for cat food is , where Q is pounds of cat food, and the owners with multiple cats have demand for cat food equal to . The marginal cost is $2.00 per pound. Compare consumer surplus and aggregate surplus with and without discrimination. Which situation maximizes surplus?
8. Suppose that soft drink manufacturer JP-Cola is conducting a 10-day test of a new type of vending machine that charges a price according to the outside temperature. On "hot" days, demand for vending machine soft drinks is. On "cool" days, demand for vending machine soft drinks is. The marginal cost is $0.20. What price should JP-Cola charge on hot days? What price should JP-Cola set on cool days? Suppose that, on average, half of the days are hot and half of the days are cool. What price should the JP-Cola set if they instead charged a single price? Compare JP-Cola's profits with and without price discrimination.
9. Suppose that soft drink manufacturer JP-Cola is still conducting a 10-day test of a new type of vending machine that charges a price according to the outside temperature. On "hot" days, demand for vending machine soft drinks is and on "cool" days, demand for vending machine soft drinks is. The marginal cost is $0.20. Compare consumer surplus and aggregate surplus with and without price discrimination. Which situation maximizes surplus?
0. Could price discrimination result in greater consumer surplus? What would cause this to occur?
1. Dolly's Pet Shop has decided to extend its dog food refill program to include cat food. There is an initial fee to purchase the tub and then customers can refill the tub at a per-pound rate. Again, Dolly has two types of customers for cat food; people with a single cat (S) and those with multiple cats (M). The 200 single-cat owners' demand for cat food is, where Q is pounds of cat food. The 100 owners with multiple cats have demand for cat food equal to. The marginal cost is $2.00 per pound. If Dolly sells to both types of customers, which of the following per-pound prices is the most profitable: $1.00, $1.50, $2.00, or $2.50? What are the associated tub (or fixed) fees for each price per pound?
2. Following from question (9), if instead of a single fee, Dolly's Pet Shop has decided to offer a pair of two-part tariffs. The plan for the 100 owners with multiple cats (where demand is) is to offer the cat food at $2.00 per pound. The plan for the 200 single cat owners (with demand) will be capped at the quantity of cat food a single cat owner decides to purchase given the per-pound price in that plan. Dolly's marginal cost is $2.00 per pound. Which of the following per-pound prices in the plan intended for single cat owners is the most profitable: $1.50, $2.00, $2.50, or $3.00? What are the associated tub (or fixed) fees for each price per pound for each type of consumer?
3. Dolly's Pet Shop has again decided to extend its dog food refill program to include cat food, but with a revised number of each customer type. There is an initial fee to purchase the tub and then customers can refill the tub at a per-pound rate. Dolly has two types of customers for cat food; people with a single cat (S) and those with multiple cats (M). The 200 single-cat owners' demand for cat food is, where Q is pounds of cat food. The 300 owners with multiple cats have demand for cat food equal to. The marginal cost is $2.00 per pound. If Dolly sells to both types of customers, which of the following per-pound prices is the most profitable: $1.00, $1.50, $2.00, or $2.50? What are the associated tub (or fixed) fees for each price per pound?
4. Following from question (11), if instead of a single fee, Dolly's Pet Shop has again decided to offer a pair of two-part tariffs. The plan for the 300 owners with multiple cats (where demand is) is to offer the cat food at $2.00 per pound. The plan for the 200 single cat owners (with demand) will be capped at the quantity of cat food a single cat owner decides to purchase given the per-pound price in that plan. Dolly's marginal cost is $2.00 per pound. Which of the following per-pound prices in the plan intended for single cat owners is the most profitable: $1.50, $2.00, $2.50, or $3.00? What are the associated tub (or fixed) fees for each price per pound for each type of consumer?
5. What is bundling? How can a firm use bundling to extract additional consumer surplus?
Chapter 19
Sample Problems
. Lea and Gayle are duopolists each owning a tourist guide service in a popular scenic town. The market demand function is, where P is the price of tours around the town's main attractions and is the number of tours demanded per year. The marginal cost is $75 per tour. Competition in the market is described by the Cournot model. What are Lea's and Gayle's equilibrium outputs? What is the price? What do they each earn in profits? How does the price compare to marginal cost? How do these profits compare to the monopoly price and profit?
Answer:
First note that inverse demand is. Given Lea's (L) and Gayle's (G) outputs,and the price is
The residual demand is,
To find the Nash equilibrium:
Find Lea's best response function by deriving her marginal revenue, then set up profit maximizing conditions.
Set MR = MC:
By the symmetrical nature of the problem we can deduce without calculation that Gayle's best-response function is. Thus the Nash equilibrium is:
Substituting Gayle's output into Lea's best-response function we find that Lea's output is also 5000. Using this information we can recover the market price:per tour. Both Lea and Gayle will each earn a profit ofper year.
Note that P = $90 is higher than the perfectly competitive price of $75. To find what the price would be if there was a monopoly in this market use the same inverse demand curve without partitioning it into residual demand curves, then find the marginal revenue and set up the profit-maximizing condition:
and set MR = MC:
Substituting this quantity we find that P = $97.5 and profit for the monopoly firm is $168,750 per year.
2. What factors affect the number of firms entering a market and what are economists referring to when they say a market has intense competition?
Answer:
There are several factors that can affect the number of firms entering a market. If the fixed costs associated with becoming active in the market decrease, firms have an incentive to enter. However, this depends on the size of the market as well. If the market is relatively small then there is a greater incentive to enter the market because it is likely that there is still room to make a positive profit.
Another factor that can affect the entry of firms into a market is what is known as intensity of competition. Economists declare competition is more intense in one market than another if given any number of firms, the equilibrium price in the first market is lower than the price in the second market.
3. When does monopolistic competition occur?
Answer:
Monopolistic competition occurs in a market characterized by free entry and a large number of firms, each of which produces a unique product (i.e. heterogeneous goods), prices are set above marginal cost, and firms earn (close to) profit net of its fixed costs.
4. The market demand function for tourist guide services is , where P is the price of tour around the town's main attractions and is the number of tours demanded per year. The marginal cost is $75 per tour. Suppose Lea enters this market first and chooses her output. How much larger is her profit compared to the situation in question (3) described by the Cournot model. How do Gayle's profits in the two cases compare?
Answer:
Recall Lea's residual demand curve from problem (1).
Recall also, Gayle's best-response function: . By substituting Gayle's best-response into Lea's inverse residual demand function, we get a formula characterizing the price Lea receives at any output she might produce, while taking into account Gayle's response to her choice:
Now derive Lea's marginal revenue:
To maximize her profit Lea will set MR = MC.
Gayle's best response is. At a total market output of 11,250 the market price is. This price implies that Lea's profit is $84,375 per year, while Gayle's profit is $42,187.50. Lea makes $9,375 more, and Gayle made $32,812.50 less, than in the case characterized by the Cournot model.
Additional Exercises for Chapter 19
. What is the Nash equilibrium in the Bertrand oligopoly model? How many firms in a single market are needed to reach this equilibrium?
2. What oligopoly model involves choosing quantities? How is this model related to the possible existence of a capacity constraint?
3. Kristin and Megan are each producers of environmentally friendly handbags in a duopoly market. The market demand function is. The marginal cost is $20 per bag. If competition in this market follows the Bertrand model, what is the equilibrium price? How many handbags are bought and sold? What do they earn in profits?
4. Kristin and Megan are each producers in a duopoly market facing the market demand function. The marginal cost is $20 per bag. If competition in this market follows the Cournot model, the corresponding best response functions are and . What is the Nash equilibrium output? What is the equilibrium price? What do they each earn as profit?
5. Kristin and Megan are duopolists in the handbag market. The market demand function is. The marginal cost is now $30 per bag. If competition in this market follows the Cournot model, what is the Nash equilibrium output? What is the equilibrium price? What do they each earn as profit?
6. Kristin and Megan are duopolists in the handbag market. The market demand function is now. The marginal cost is $10 per bag. If competition in this market follows the Cournot model, what is the Nash equilibrium output? What is the equilibrium price? What do they each earn as profit?
7. Suppose a small town is served by two competing hamburger stands, Joe's Burgers and Burgers by Bob, who face the daily demand functions and. The marginal cost is $1.00. What are the Nash equilibrium prices when the two firms set their prices simultaneously? What are the quantities sold by each stand and what are their resulting profits?
8. Two competing hamburger stands, Joe's Burgers and Burgers by Bob, face the daily demand functions and . The marginal cost is $1.00. What are the Nash equilibrium prices when the two firms set their prices simultaneously? What are the quantities sold by each stand and what are their resulting profits?
9. Assume that Kristin and Megan each sell designer handbags and share the monthly collusive profit of $10,000 equally, $5,000 each. If Kristin undercuts Megan, she will earn an additional $5,000 that month, but Megan will immediately cut her price to marginal cost in all future months. What interest rate will make this cooperative arrangement supportable?
0. Again assume that Kristin and Megan each sell designer handbags, but now they share the monthly collusive profit of $16,000 equally, $8,000 each. If Kristin undercuts Megan, she will earn an additional $8,000 that month, but Megan will immediately cut her price to marginal cost in all future months. What interest rate will make this cooperative arrangement supportable?
1. What is the difference between explicit collusion and tacit collusion? Why is a tacitly collusive agreement harder to maintain?
2. What is business stealing and how might this lead to excessive entry into a market?
3. Kristin and Megan are each producers of environmentally friendly handbags in a duopoly market facing the market demand function. The marginal cost is $20 per bag. Suppose Kristin enters this market first and chooses her output before Megan. What is Kristin's profit-maximizing output? What is her profit at that output? What is Megan's profit-maximizing output in response? What are Megan's profits? Finally, compare Kristin's profit in this case to her profit when quantities were chosen simultaneously (in question #4).
4. Kristin and Megan are each producers of environmentally friendly handbags in a duopoly market facing the market demand function. The marginal cost is $10 per bag. Suppose Kristin enters this market first and chooses her output before Megan. What is Kristin's profit-maximizing output? What is her profit at that output? What is Megan's profit-maximizing output in response? What are Megan's profits? Finally, compare Kristin's profit in this case to her profit when quantities were chosen simultaneously (in question #6).
5. What is a horizontal merger and why would this be more effective than fixing prices?
Chapter 20
Sample Problems
. What does Coase's theorem say? Why might the assumptions of frictionless bargaining be unreasonable?
Answer:
Coase's theorem says that if bargaining is frictionless, then it does not matter how property rights are assigned and voluntary agreements between private parties will remedy the market failures associated with externalities and restore economic efficiency.
Assuming that bargaining is frictionless is a critical assumption that is often made unreasonable by the fact that the act of bargaining is rarely frictionless. Bargaining is often impractical, requiring time and effort as well as money if lawyers become necessary. Aside from these setbacks of bargaining, parties that are negotiating with one another have limited information about the others costs and benefits. Another feature that slows down bargaining is the difficulty encountered when trying to enforce these contracts.
2. What is a tradable emissions permit, and why would such a permit promote least-cost abatement?
Answer:
A tradable emissions permit gives a firm the right to pollute a given amount of a specified pollutant. This permit is transferable, meaning that one firm may sell it to another firm if they wish.
Emissions permits promote least cost abatement because they are easily transferable. A firm with low marginal cost of abatement can sell some of their permits to any firm with higher marginal cost of abatement. This puts imposes a market system structure on pollution which forces firms to take pollution generation into account when maximizing production and ultimately profits.
3. You live by the local airport which until recently only allowed single engine propeller planes. In an effort to increase revenue, the airport signed a deal that will allow private jets to land at the airport. Now you and your neighbors are dealing with the constant nuisance of 'take-off' noise. Let D stand for the noise level of planes taking off in decibels, B is the benefit to the airport, and C represents your costs. For any given volume of noise the local airport's benefit iswhich means that the airports marginal benefit is. Your cost is , which means your marginal cost is. Given these costs and benefits, what is the efficient noise limit? What is the efficient Pigouvian tax?
Answer:
Set MB = MC:
The efficient noise limit is 90.2 decibels. At this socially efficient level your marginal cost is. Therefore, the efficient Pigouvian tax is $1.45 per decibel. With such a tax the airport will choose 90.2 decibels of noise, paying a total of $130.79.
4. What would the government need to do to operate a Groves mechanism? Would a citizen have an incentive to over-represent his/her need of a public good?
Answer:
The government would have to ask each citizen to report the total benefit he/she would receive from the public good at any possible level of provision, so that it could calculate every individual's marginal benefit. The government would take the information given by each citizen as a true representation of their benefit.
A Groves mechanism is designed in such a way that citizens would have no incentive to exaggerate their need of a public good. If a citizen were to exaggerate her need for a public good, she would be faced with a higher mandatory contribution to the public good, which would not be proportional to her actual preferences.
Additional Exercises for Chapter 20
. What is a negative production externality and how does it create a deadweight loss?
2. The market demand function for down blankets isand the market supply function is, as seen in the graph below. Suppose that the production of down blankets creates an external cost per blanket equal to $50. Illustrate the social cost curve on the graph below and shade in the deadweight loss from the external cost.
3. Again, the market demand function for down blankets isand the market supply function is, as seen in the graph below. Suppose that the production of down blankets creates a marginal external cost . Illustrate the social cost curve on the graph below and shade in the deadweight loss from the external cost.
4. Assume that 10 firms compete in the construction market, which produces silt that pollutes the water supply. The total cost to each firm is , where Q is the monthly number of construction sites in progress. The marginal cost is therefore. The external cost of the silt runoff is , making the marginal external cost . The monthly demand for construction is. First solve for the competitive equilibrium price and quantity. Next solve for the socially efficient quantity. Finally, calculate the deadweight loss resulting from the negative externality.
5. Again assume that 10 firms compete in the construction market, which produces silt that pollutes the water supply. The total cost to each firm is , where Q is the monthly number of construction sites in progress. The marginal cost is therefore. The external cost of the silt runoff is , making the marginal external cost . The monthly demand for construction is. First solve for the competitive equilibrium price and quantity. Next solve for the socially efficient quantity. Finally, calculate the deadweight loss resulting from the negative externality.
6. Assume that 10 firms compete in the post-construction beautification market, which re-plants trees that had been cut down for the construction. The total cost to each firm is, where Q is the monthly number of construction sites in progress. The marginal cost is therefore. The monthly marginal benefit to owners of the land is. However, other people who drive down the street also benefit from the beautiful trees, creating an external benefit , making the marginal external benefit. First solve for the competitive equilibrium price and quantity. Next solve for the socially efficient quantity. Finally, calculate the deadweight loss resulting from the positive externality.
7. What are emissions standards? Why might both sides (both polluter and pollutee) have an incentive to exaggerate their costs?
8. What is a Pigouvian tax and what must this tax be set equal to in order to achieve the socially efficient outcome?
9. Your neighbor across the street loves to collect garden gnomes and has them scattered all across her lawn. They are such an eyesore that you and the other surrounding neighbors have determined your cost (in dollars) to be , with a marginal cost of , where Q is the number of garden gnomes. However, your neighbor loves her gnome collection and her benefit is, making her marginal benefit. Given this information, what is the efficient number of gnomes? If the neighborhood association sought to issue a Pigouvian tax, what is the efficient tax per gnome?
0. Again, your neighbor across the street loves to collect garden gnomes and has them scattered all across her lawn. They are such an eyesore that you and the other surrounding neighbors have determined your cost (in dollars) to be , with a marginal cost of , where Q is the number of garden gnomes. Suppose the neighborhood association is considering weighing cost and benefit to determine the efficient number of gnomes and, when asked, your gnome-loving neighbor claims her benefit to be, making her marginal benefit. How does her new "estimate" of her benefit change the efficient number of gnomes? Why might she have an incentive to overestimate her benefit?
1. How does a liability rule force polluters to internalize their external costs? What other costs may be associated with the enforcement of such a rule?
2. How is a public good defined and what is the free-rider problem?
3. Suppose that some students value having potted plants in the classroom and believe that plants help them relax and learn better. If there are 7 of these students in the class and each has a marginal benefit, where Q is the number of potted plants. If the marginal cost per plant is $14, how many plants would each student choose individually? What is the socially efficient number of plants?
4. Suppose that some students value having potted plants in the classroom and believe that plants help them relax and learn better. If there are 5 of these students in the class and each has a marginal benefit, where Q is the number of potted plants. If the marginal cost per plant is $10, how many plants would each student choose individually? What is the socially efficient number of plants?
5. What is the median voter theorem? What does this theorem require to result in a socially efficient outcome?
Selected Solutions
Chapter 1
. A capitalist economy is more closely described as a free market system. In a free market system, the government usually lets markets operate freely with fewer regulations. In contrast to communist economies, capitalist economies are characterized by having a higher degree of decentralization, with the means of production primarily owned and controlled for the profit of private individuals.
3. The principle of individual sovereignty enables economists to avoid paternalistic judgments. Applying the principle of individual sovereignty allows economists to turn a positive analysis into a normative analysis because invoking the principle of individual sovereignty allows economists to assume individuals have full knowledge of the consequences of their actions.
5. The best theories are ones that are generally applicable and specific in their implications. Theories that identify additional implications are paramount to developing a broadly applicable theory. In essence, a theory should try and explain as many things as it can without explaining things that it can't.
7. The three main categories of economic data are: Surveys, Records, and Experiments. Surveys are a method of collecting data that economists use because they can provide data on virtually any subject. Economists' main obtain survey data for a fee or free of charge in many cases. Records on financial accounts, personal records, and customer data, are typically kept by companies for their own research and analysis. Some of these records are open to the public, particularly for publicly traded companies, and they may be obtained for a fee or free of charge.
9. In a controlled experiment, economists isolate causal relationships by design, whereby they typically have control and treatment groups. A natural experiment differs from a controlled experiment primarily because they occur naturally without design but they produce similar effects. Natural experiments create a situation where by chance two otherwise identical groups may differ, creating the same effect as separating subjects into controlled and treatment groups.
1. Trade can benefit two people at the same time for several reasons. Trade allows people to exchange something they own that is of relatively lesser value, for a good or service that is of relatively greater value. Trade also allows workers, countries, companies, etc. to specialize in producing goods and or services they produce well.
Chapter 2
.
The function is:
Y-intercept (Price) = 13.5 & X-intercept (Quantity) = 27
3. The demand curve shifts to the right.
5.
The Y-intercept (Price) = 1.83
7. In the answer to (6) we see that as the price of potato flakes decreases, the quantity of French fries supplied increases, demonstrating that the two goods are substitutes in production.
9. ; million lbs.; 8 million lbs.
1. ; million lbs.; 10 million lbs.
3. The equilibrium price for the market is $4.00. At this price 8 gallons of ice cream are bought and 8 gallons of ice cream are sold. See the graph below.
5. Elasticity of supply at:
Elasticity of supply at:
Chapter 3
. The opportunity cost of going to see the movie is not being able to go to the gym. Working out at the gym represents your opportunity cost in this case because it is your next best alternative according to your preference rakings. If going to see a movie was not an option and you had chosen to go to the gym, your opportunity cost would be time not spent reading your book.
3.
Hours
Benefit
Cost
Net Benefit
0
0
0
0
460
80
380
2
840
220
620
3
140
420
720
4
360
680
680
5
500
000
500
6
560
380
80
The best possible option here is the one with the highest net benefit. This would be where you employ your mechanic for 3 hours and your net benefit is $720.
5.
Hours
Benefit
Marginal Benefit
Cost
Marginal Cost
460
460
80
80
2
840
380
220
140
3
140
300
420
200
4
360
220
680
260
5
500
40
000
320
6
560
60
380
380
The best choice according to this schedule is to hire your mechanic for three hours. The three hour mark is the last point at which the marginal benefit exceeds the marginal cost. This is in accordance with the principle of No Marginal Improvement, which states that "at a best choice, the marginal benefit of the last unit must be at least as large as the marginal cost, and the marginal benefit of the next unit must be no greater than the marginal cost."
7. The graph below provides us with similar information as charts from previous problems. To find the best choice we merely have to look for the largest vertical distance between the total benefit and total cost curves (Note: the only relevant portion of the graph is where TB > TC, because points that lie where TC < TB are clearly not best choices) this strategy will ensure that we have identified the largest net benefit.
9. According to the chart below, hiring your mechanic for 4 hours is the best choice, because it has the largest net benefit of any of the possible choices.
Hours
Benefit
Cost
Net Benefit
0
0
0
0
335
65
300
2
640
80
460
3
915
345
570
4
160
560
600
5
375
825
550
6
560
140
420
1. Sunk costs should be ignored when finding the best choice from a set of options because for one they are unavoidable. More importantly, sunk costs have no effect on marginal costs.
3. The best choice is found by setting MB = MC (by No Marginal Improvement Principle):
=
hours.
If the mechanic requires a 1-hour minimum payment then it would be better not to take your car to the mechanic (choose H = 0). By the No Marginal Improvement Principle, if you increased the units of H to 1 hour MB < MC and thus the net benefit will be negative.
5. The choice you make will depend on which option (4 or 5 hours) gives you a higher net benefit. If the MB grows smaller and MC grows larger as the number of hours increase, then 4 hours would be a better choice.
Chapter 4
. It is important that consumers follow the choice principle because it allows them to select, among available alternatives, which one ranks highest. If the choice principle did not apply we could not assume when formulating economic theories that consumers are rational. Assuming consumers are rational is an important assumption economists make when they model economic phenomena, as we saw from chapter 1.
3. a) No, she would swap b) No, she will not swap c) No, she will not swap
5. D is most preferred (by the more-is-better principle). Both B and E are equally preferred because they are on the same indifference curve (2nd most preferred). 3rd most preferred is C, and the least preferred bundle is A. This may also be represented in the following form ().
7. This type of preference is called perfect complements.
9.
The graphs shows that given 2 pieces of cake Ashleigh could not be made better off by giving her any amount of pie. Conversely, the graph also shows that given 2 pieces of pie Taylor could not be made better off by giving her any amount of cake.
1. Given Stephanie's utility function the formula for her indifference curves can be found by rearranging the equation and solving for IC, as such: .
Stephanie's marginal utility of pie is (addingincreases the utility value by 2P so, The marginal utility of ice cream is (adding ?IC the utility value by 2 so, Now, to find Stephanie's marginal rate of substation for pie with ice cream use the equations for finding the MRS with the information about the marginal utilities of both pie and ice cream, as such:
Her MRS of pie with ice cream is P scoops of ice cream per piece of pie.
3. Finding the indifference curves is simply a matter of solving the utility function given in terms of P like so (note: for simplicity U(P,IC) = U):
To find Dave's marginal utility of pies:
Finding Dave's marginal utility of ice cream can be done in the same fashion:
Finding the MRS of pie with ice cream can be done as follows:
5. There is certainly a trade that could improve both of their positions. This assertion can be verified as follows: if Stephanie has 4 scoops of ice cream her utility function is: If Dave has 3 pieces of pie his utility function is: . If Dave gives Stephanie 1 piece of pie for 1 scoop of ice cream, Stephanie's utility would increase to Dave's utility would also increase to , therefore this trade is beneficial to both Stephanie and Dave.
Chapter 5
.
Pie (pieces)
3
6
7
8
9
2
4
5
6
7
2
3
4
5
0
0
2
3
0
2
3
Ice Cream
(scoops)
If Stephanie's income is $4.00 her affordable options are lightly shaded. If Stephanie's income is increased to $6.00 her affordable options increase to include the lightly shaded regions and the more darkly shaded regions.
3. If the price of ice cream doubles to $2.00 the new budget line is. Her new budget line has a slope of -1, with a horizontal intercept at 2 scoops of ice cream, and a vertical intercept at 2 pieces of pie.
The number of affordable options has decreased due to the increase in the price of ice cream scoops. This increase in the price of ice cream has caused the original budget line to rotate inward from the origin (pivoting at the intercept for pie).
5. The unaffordable bundles are lightly shaded below. Of the affordable bundles, the one that would give Stephanie the most utility is the bundle with 2 scoops of ice cream and 1 piece of pie. This bundle corresponds to her highest ranking (8) affordable option.
Pie (pieces)
3
0
5
2
2
2
7
4
3
4
9
8
6
0
5
4
3
1
0
2
3
Ice Cream
(scoops)
7. The introduction of pie rationing forces to Stephanie to forgo her otherwise affordable and highest ranked option (#2: 3 pieces of pie, 2 scoops ice cream), and accept her next best option (#3: 2 pieces of pie, 3 scoops of ice cream).
9. It is important for the budget line to be tangent to the indifference curve at a utility-maximizing bundle for interior solutions because an indifference curve going through any interior choice that not satisfy the tangency condition thus creating an overlap between the areas below the budget line and above the indifference curve. Boundary solutions are excluded from the tangency condition because at a boundary solution the slope of the budget line and the slope of the indifference curve are not equal.
1. Use the tangency condition to solve for B in terms of C:
Now use the previous result and the budget line to solve for C and B:
Solving for B:
With this marginal rate of substitution Holly should buy 3.33 pounds of candy and 3.33 packages of balloons.
3. First, substitute,, andinto the formula.
Next, solve for B and C using the budget line.
Solving for C:
Therefore, Holly will buy 1.5 packages of balloons and 7 pounds of candy for her classroom.
5. To calculate the optimal amounts of candy and balloons at $5 and $10 set up the budget lines at the different prices of balloons:
If balloons cost $5:
Next set up the following:
Use the budget line to solve for B and C:
If balloons cost $10:
Next set up the following:
Use the budget line to solve for B and C:
Use the information above to plot the graphs for the budget lines, optimal solutions, and finally the price-consumption curve.
7. To plot the Income-consumption curves we first need to find the budget lines. Since the prices of balloons and candy are constant and the only thing changing is income it is fairly simple to find the budget lines:
If:()
If:()
If:()
Next, we need to find the optimal bundles for each budget line. To do this we use the following equation:
Using the previous result you can solve for the optimal bundles at each budget line.
The optimal bundles are as follows:
For:
For:
For:
Now we have all of the information needed to plot the income-consumption curves and the Engel curves.
Chapter 6
. The difference between the substitution effect and income effect of a price change is the substitution effect represents a change in relative prices that causes a consumer to substitute one good for another. On the other hand, the income effect represents how price change affects a consumer's purchasing power. The effect of an uncompensated price change comes in two parts: (1) the effect of a compensated price change and (2) the effect of removing the compensation. In relation to the uncompensated price change the substitution effect represents part (1) and the income effect represents part (2).
3. To find the uncompensated effect of Holly's purchases of candy and balloons first use the tangency condition to solve for the B and C using a generic form of a budget constraint as follows:
(Budget line)
Tangency condition:
Now use the previous result and the generic budget line to solve for B and C:
Now, for C:
So, withshe chooses. If the price of balloons falls to $5.00 she chooses.This uncompensated price reduction raises Holly's balloon package purchases from 3 to 8 and leaves the amount of candy she buys unchanged.
Next, to find the compensated effect we need to used the tangency condition and find the relationship between candy and balloons at their new prices.
Ifthe relationship between candy and balloons is:
Holly's best choice with a compensated price change also lies on the indifference curve that passes through the bundle. Using the formula for her indifference curvethe value of U that runs through the bundleis 50, which implies that Holly's formula for this indifference curve is:
A bundle can satisfy both the tangency condition formula and the indifference curve formula only if: Using the quadratic formula yields
Next, using the tangency condition, we get that. So the compensated price effect shifts Holly from the bundleto the bundle. (Note: you can check to see that both bundles are on the same indifference curve by plugging in each bundle into the utility function; both bundles should have the same utility).
At the new prices, Holly needs $60.70 to buy the bundle. Since her income is $50, we would have to give her $10.70 to compensate for the price cut.
To break up the uncompensated price change into substitution effects and income effects:
Note that the substitution effect is the same as the effect of the compensated price change. The income effect is what is known as the residual; it shifts Holly from the bundleto the bundle.
5.
The reduction of the price of good Y causes the rotation ofto. Bundle A shown on the graph above is the bundle chosen before the price decrease of good Y. The arrow from A to C on the graph above shows the substitution effect of the change in the price of good Y. Bundle B on the graph above is the bundle chosen after the price decrease of good Y. The arrow from C to B on the graph depicts the income effect of the price decrease; and bundle C represents the bundle chosen on the compensated budget line after the price change putting the consumer back on their original indifference curve.
7. A Giffen good is a good that's quantity sold increases as its price increases. The law of demand says that for a normal good as the price of the good increases the quantity sold decreases. So, a Giffen good would violate the law of demand.
9. First we must use the budget line and the tangency condition to solve for optimal values of candy and balloons.
(Budget line)
Tangency condition:
Solving for B and C we get&
Next, use the utility function in its functional form to solve for U with the optimal bundle:
Holly's indifference curve formula through this bundle is:
If the store does not have balloons,Therefore, to place Holly on the same indifference curve, we must pick the value of B for whichSolving for C, we getTo buy 12.25lbs of candy Holly would need $61.25, so the store would have to pay her $11.25 to compensate for not having balloons in stock. This is Holly's compensation variation.
1. The horizontally shaded triangle represents Holly's consumer surplus before the price change, when the price of paint was $6. Here the consumer surplus is:
The number 3.2 is found by evaluating the quantity sold at $6 using the demand function.
If the price of paint increases to $7 then Holly's consumer surplus decreases, as is depicted on the graph below by the entire shaded triangle. You can calculate the change in the consumer surplus as follows:
Take the area of the triangle before the price change and subtract it by the area of the triangle after the price change.
The number 2.4 is found by evaluating the quantity sold at $7 using the demand function.
3. An increase in the wage rate could increase the amount of labor supplied if the increase in the wage rate increases the cost of leisure to a point where the laborer is not willing to substitute leisure for less labor supplied.
An increase in the wage rate could decrease the amount of labor supplied if the increase in the wage rate increases the purchasing power of a laborer sufficiently enough to offset the increase in the cost of leisure to the point where leisure time rises at the expense of labor supplied.
Both of these situations are possible with a backward bending supply curve.
5. To find the equation of the compensated demand curve first take the formula for the tangency condition and the price of balloons equal to 10:
Next, rewrite the utility function as follows using the bundle given:
Now set the tangency condition formula equal to the indifference curve formula equal to each other and solve for C:
This is the compensated demand curve.
Chapter 7
. Use the graph to see which methods are inefficient.
Production
Method
Units of Labor
Output
Efficient?
A
48
Yes
B
2
24
Yes
C
2
12
No
D
3
205
No
E
3
216
Yes
F
4
312
Yes
G
4
295
No
Production methods that are inefficient are characterized by lower levels of output per unit of labor (or any other input).
3. To find how many cups of lemonade Giselle can produce daily with one worker, use the production function provided, and evaluate it where L = 1:
cups of lemonade
If another worker is hired, then L = 2:
cups of lemonade
These answers show that at low levels of inputs the marginal product of labor is increasing at an increasing rate.
5.
if L = 0
if L = 1
if L = 2
if L = 3
if L = 4
if L = 5
7.
MPL APL
if L = 0:
if L = 1: 42 42
if L = 2: 74 58
if L = 3: 88 68
if L = 4: 84 72
if L = 5: 62 70
9. Notice that as the number of workers increases the MPL increases to a certain point, and then begins to decrease.
Units of Labor
Output
0
0
0
0
48
48
48
2
24
76
62
3
216
92
72
4
312
96
78
5
400
88
80
6
468
68
78
1.
Units of Labor
Units of Capital
Output
48.00
2
57.08
2
80.73
2
2
96.00
2
3
06.24
3
2
30.12
3
3
36.30
Looking at output levels associated with either an increase in a unit of labor, or a unit of capital we see that labor is more productive than capital at Pucker Up.
3. This family of isoquants demonstrates that jelly and peanut butter are perfect complements. The graph shows for example, at 2 oz. of jelly the firm cannot be made better by having more peanut butter while jelly is at 2 oz.
5. Giselle's MRTS remains unchanged. This type of productivity improvement is called a factor-neutral technical change, which is characterized by having no affect on the MRTS at any output combination. It only changes the output level associated with each of the firm's isoquants.
Chapter 8
. The key characteristic that distinguishes an avoidable fixed costs from a sunk fixed cost is that with the former a firm doesn't incur the cost (or can recuperate it) if it produces not output. A sunk fixed cost is incurred even if the firm decides not to operate.
3. First, solve for the number of hours of labor needed to produce Q cups of lemonade:
(take the square root of both sides)
Next, multiply the wage rate by the previous result to get the variable cost function . The short-run cost function is therefore .
5. We know the MRTSLK for the Cobb-Douglas production function is which has a declining MRTSLK.
Now set the MRTSLK equal to the input price ratio: where , , w = $80 per day, and r = $40.
The tangency condition holds at the point on the 150 cup per day isoquant where the capital-labor ratio equals 2. When Giselle uses L units of labor and 2L units of capital, the output is:
Producing 150 cups of lemonade requires:
workers
Because we rounded to 7 workers, the amount of capital (K) needed is 2L.
2(7) = 14 units of capital
So Giselle's total cost for one day is $1120.
7. In problem 5 we found that to produce Q cups of lemonade Giselle needs the amount of labor L that solves the formula: and she needs 2 times the amount of capital. Thus Giselle needs workers and units of capital. The cost of producing Q units is therefore,
Similarly, in problem (6) Giselle's formula for Q cups of lemonade, and L amounts of labor: and she needs the amount of capital. Thus, Giselle needs workers and units of capital.
The cost of producing Q units is therefore,
9. The relationship between Giselle's labor input and her output of lemonade is the same as was worked out in problem 3. So her cost function is
1. We know that with a least-cost plan that assigns a positive amount of output to each stand, marginal cost must be the same at both stands. Thus Giselle needs to divide production between her two stands so that their marginal cost are equal. Doing this means choosingandso that MC1=MC2. Since total output equals 225 cups of lemonade, we knowso we can express MC2 as follows: .
Next, to find how much each stand will produce, set MC1=MC2:
Therefore, Giselle must produce 75 cups of lemonade in stand 1 and 150 cups in stand 2. Her total cost of production will be:
3. The fixed cost for the firm is $100. This number might represent a sunk cost, like the cost to rent a building. A fixed cost will not depend on how much output is produced.
To identify firms variable cost V(Q) in the function, look at the parts of the equation that are dependent on Q. IT follows that the .
To find the average cost (AC) use the fact that . In this firm's case it is:
To find the AVC and AFC use the following fact:
Now, . Thus, and .
5. If Giselle's capital is fixed at 10 units, to produce Q cups of lemonade the amount of labor that solves the formula is:
So her short-run cost function is,
Giselle's long-run cost function is, (as was found in problem 7).
Note that Giselle's short-run and long-run costs are equal when Q=150. (Since
Chapter 9
. We can find the inverse demand function by solving
for P, which gives us
To determine the price at which, solve for P such that
Megan should price each shirt at $20.
If, solve for P such that,
Megan should price each shirt at $15.
3. A price-taking firm cannot affect the market price for a good by any means. A price-taking firm cannot change the price by selling their good at a higher or lower price than the market price. Therefore, no matter how many units of a good a price-taking firm sells they will always have the same price leading to the characteristic horizontal demand curve.
5. Use the quantity rule. Thus, Megan's best positive sales quantity solves the formula , or
Now we must check the shut-down rule, by calculating the profit from producing 108 T-shirts.
Megan should go ahead and produce 108 T-shirts. If Megan also has an avoidable cost of $200 per day her profit at Q = 108 would be negative, and the shut-down rule would tell Megan to stop producing T-shirts all together.
7. The best positive sales quantity, Q, is found when the price P is set equal to the marginal cost, so that
Solving for Q, we find that
If Megan has no avoidable fixed cost, . Note: , so. Since this is an increasing function we know that $3 is the minimum. So using the shut-down rule, Megan's supply function is
When we add an avoidable fixed cost of $200 per day, this does not change MR or MC, but it does change when Megan wants to stay in business. We need to determine the new level of. This is done by setting AC = MC to determine the new efficient scales of production.
Solving for Q, we find. Substituting this quantity into the equation for average cost shows us that
So Megan's supply function is now
9. The minimum price at which Megan will choose to sell is $5.
1. Let's begin with Megan's long-run supply function first. If the price is greater than 150, we have P > MC at every output level. Because they make a positive profit on every unit they sell, they will want to sell an infinite amount. If P < 150, they will want to sell Q = 0. If , the firm will be willing to supply any amount, since they earn zero profit no matter how much they sell.
For Megan's short-run supply curve, we must apply the quantity rule. The quantity at which P = MC solves the formula P = Q. Thus, the most profitable quantity given the price P is Q = P. At this quantity Megan's revenue isand her avoidable costs are and she will produce in the short run as long as P > 0. The short run supply function is therefore
3. Before the avoidable cost this firm's producer surplus is:
With an avoidable fixed cost this firm's producer surplus is the trapezoidal region A, or equivalently:
5. First, start by applying the quantity rule to find the sales quantities for both the shirt and the hat, where P = MC.
(1)
and,
(2)
Now with equations (1) and (2) we can solve forand. This can be done in several ways but we will proceed as follows:
Rewrite (1) and (2).
(1)
(2)
Substitute (1) into (2) and solve for.
Solve for.
Megan's profit is:
Applying the shut down rule, we have to compare this profit not only to Megan's profit if she would shut-down her operation, but also, her profit if she would shut-down the production of either of her products, and only would produce one product.
If Megan is not producing her T-shirts such that, her cost function becomesand. The best quantity of hats is, which yields $16 profit which is less than what Megan makes if she produces both products.
This was found by setting P = MC:
Megan's profit is,
If Megan stops producing the hats so that, her cost function becomesand. Set P = MC:
Megan's profit is,
Again, this is less than Megan would make if she produced both the T-shirts and the hats, so Megan should produce the profit-maximizing sales quantities ofand.
Chapter 10
. Compounding refers to the payment of interest on a loan balance that includes interest earned in the past. Compounding causes the loan balance to grow faster as time passes. It is important to factor compounding interest when you are making a saving decision because you should realize that you won't see immediate large returns on your investment, but they will increase over time. If you are about to make a borrowing decision it is important to factor compounding because as a borrower you should understand that you will have a larger loan balance as more and more time passes.
3. To calculate the PVD of $100 received in 5 years at the various interest rates shown, we need to identify some variables from the formula:
Notice that, T, tells use in years how long it will take to received the future value F, which is $100, and R, is the interest rate which will vary in this problem:
If the R = 5%:
If the R = 10%:
If the R = 20%:
5. To find out how much you would owe your friend in today's dollars, use the formula for finding the present discounted value of a stream:
Therefore,
So in today's dollars you will have to pay your friend $1005.46 to borrow $1000.
7. Option (1) requires no calculations; it is simply $25,000 in today's dollars.
To figure out how much option (2) will cost in today's dollars we must perform the PDV calculation for a constant stream:
To calculate how much option (3) will cost in today's dollars we must perform the PDV calculation for a constant stream:
Therefore, with an interest rate of 8% option (2) is the lowest price option measuring in today's dollars.
9. To find the real interest rate, use the formula:
If the rate of inflation grew to 3.5%, then the real interest rate would be:
As the rate of inflation grows the real interest rate decreases.
1. Draw the 45° line on the graph. In relation to the 45° line Brian's preference is above the line according to his indifference curve. This tells us that Brian prefers to spend more money next year than this year in order to maximize his preferences.
3. Start by identifying bundles that satisfy the tangency condition. The slope of Brian's indifference curve is(his MRS times negative one); and the slope of his budget line is. Brian's best choice is where the slope of his indifference curve is equal (tangent) to his budget line, which is, or equivalently. This tangency condition informs us that Brian consumestimes as much food this year as next year.
The formula for Brian's budget line is. To find the bundle to satisfy the tangency condition and the budget line formula, substitute forin the budget line formula using the tangency condition formula.
So, this year Brian will spend $1200 and save $1200; next year he will spend. Because the interest rate is 5%, Brian consumes 1200 pounds of food this year and 1142.85 pounds next year.
If Brian makes $2400 the first year and $1200 the next year, then Brian's new budget line is as follows:
So, this year Brian will spendand saves;next year he will spend. Because the interest rate is 5%, Brian consumes 1771.43 pounds of food this year, and 1687.07 pounds of food next year.
The formula for Brian's savings can be written as S = 1200, because when Brian makes $2400 in the first year and nothing the second year, his savings do not depend on the interest rate.
The formula for Brian's savings for the situation in which he makes money in both years is the following:
In this instance we see that Brian's savings rate depends on the interest rate.
5. Start by identifying bundles that satisfy the tangency condition. The slope of Brian's indifference curve is(his MRS times negative one); and the slope of his budget line is. Brian's best choice is where the slope of his indifference curve is equal (tangent) to his budget line, which is, or equivalently. This tangency condition informs us that Brian consumestimes as much food this year as next year.
The formula for Brian's budget line is. To find the bundle to satisfy the tangency condition and the budget line formula, substitute forin the budget line formula using the tangency condition formula.
So, this year Brian will spendand save; next year he will spend.
Because the interest rate is 5%, Brian consumes 1968.25 pounds of food this year and 786.09 pounds next year.
Brian's savings function is:
Chapter 11
. To calculate your expected payoff, set up the payoff's as follows:
$0 is your expected payoff. Flipping a coin 10 times will not change your total expected payoff.
3. To derive the constant expected consumption line, set up the expected consumption equation with EC = $400, and proceed to solve forin terms ofand.
Solving for:
If:
The slope of the line is negative one.
If:
The slope of the line is negative one-half.
5. Michael's expected consumption is,. To calculate the certainty equivalent, find the level of guaranteed consumption, B, that places Michael on the same IC curve as the risky bundle.
So the certainty equivalent is $110.25. Since, the risk premium is $6.75.
7. Michael's expected utility function is,
Using this formula, plug-in the pointto find that. So points on the indifference curve through the pointsatisfy the formula. Now, solve for.
The constant expected consumption line that runs through the pointsatisfies the formula, which can be rewritten as. Notice that the indifference curve is the same as the constant expected consumption line at all points including the point. Michael is therefore risk neutral.
9. The benefit for the high state is represented by point A. The benefit for the low state is represented by point B.
The expected benefit is. The is shown on the graph, and as you can see it is characteristically higher than the expected benefit.
The risk premium and certainty equivalent are also shown on the graph.
1. Michael's available choices lie on the budget line. In this case R = 0.4, so the slope of the budget line is -l.5. The formula for Michael's budget line is thuswhere C is a constant. Michael's budget line must pass through the point,, we know thatso C =550.
Michael's best choice is the point where his budget line and indifference curves are tangent to each other. At any best choice point, the slopes of the budget line and the indifference curve must be the same. Therefore,
Using this formula, then solve forwhich gives you,.
Now we can use bothandto find Michael's best choice, by solving algebraically.
Michael spends $314.29 if he wins. Since Michael started with $324 he did buy insurance. He spent $9.71 on insurance. This means that Michael's insurance benefit must be $24.28 (since ). If Michael gets the low state, he spends.
After purchasing insurance, Michael's risky bundle is. Points on the indifference curve that runs through this bundle satisfy the formula. To find a risk-less bundle on this indifference curve (where), we solve:
B = $240.56 is the certainty equivalent of Michael's risky bundle after purchasing insurance.
Michael's initial bundle is. This bundle satisfies the formula. Like before we need to find the risk-less bundle on this indifference curve, by solving:
$240.25 represents the certainty equivalent of Michael's initial risky bundle. The value of the insurance is 31 cents, since.
3. In practice, diversification involves undertaking many risky activities, each on a small scale, whereas hedging refers to the undertaking of two risky activities with negatively correlated financial payoffs. Hedging is a case in which diversification is particularly effective at reducing risk. Diversification can be used to mitigate risk because with diversification you would invest in many activities on small scale that are uncorrelated so that the probability of having all your activities perform poorly is small. Hedging helps reduce risk because you are essentially betting big on the only two possible outcomes so that you will always have invested on the winning outcome. Therefore, you would lose from the one investment, but ideally your other investment would offset your loss.
5. As we saw in (14) if Michael keeps the $400, he is guaranteed a benefit of 20, and a certainty equivalent of $400. Investing $400 in the company will produce an expected benefit of,
Michael's certainty equivalent is the amount X that solves, so X = 271.92. So, Michael's best choice is to keep his money, and not invest. Any investment that is under $400 will improve the investment option but not enough to change the best choice of not investing.
Chapter 12
.
PLAYER A
STAY
SWERVE
PLAYER B
STAY
SWERVE
3.
PLAYER A
LEFT
RIGHT
PLAYER B
UP
DOWN
Player B has a dominant strategy of choosing "up", because regardless of player A's decision player B's best choice is still "up". Player A does not have a dominant strategy because his/her best strategy depends upon what player B chooses.
5. Regardless of whether player A chooses "left or right", player B's best response is to choose up. If player B's best response is "up" then player A's best response is "left." Therefore the Nash Equilibrium is (up, left).
7. If Nina works X hours, then Margaret's best response can be found by setting her marginal benefit equal to her marginal cost:. Solving for Y, produces the formula that describes the relationship between Nina's choices and Margaret's best responses:
Now, if Margaret works Y hours, then Nina's best response can be found in a similar fashion. Set marginal benefit equal to marginal cost, then solve for X:
In order to find the Nash equilibrium, we need to find the values of X and Y that satisfy both best response formulas at the same time. To do this, substitute the first formula into the second formula:
Thus, we find that X = 6. Plugging this value for X into the first formula, we find that Y = 6. So Nash equilibrium in this situation has both Nina and Margaret working for 6 hours.
There are two more preferred outcomes for Nina and Margaret. The most preferred outcome would have been for both Margaret and Nina to work 8 hours because each would have received a net benefit of 384 which can be found by calculating the benefit and subtracting the cost (). The net benefit for 6 hours of work is 360 which is less than the 384 net benefit of 8 hours and the net benefit if both work 7 hours which is 378.
9. A mixed strategy equilibrium is a strategy in which each player chooses a mixed strategy that is a best response to the mixed strategy chosen by the others. Mixed strategies are best used in multi-staged games.
1. For pure strategies there are no Nash equilibria.
For mixed strategies, in equilibrium a player will not choose a dominated strategy as a best response. In the matrix, every choice can be dominated by the other player's response, therefore Nina and Margaret must randomize their choices. Let Q represent the probability that Nina holds up one finger. Let P represent the probability that Margaret holds up one finger. Nina's expected payoff from holding up one finger is, while her expected payoff from holding up two fingers is.
Since Nina's choice of P has to make Margaret indifferent between holding one finger or two fingers (so that she's willing to randomize between the two), we know that, which means that. Margaret's expected payoffs are for one finger andfor two fingers; and by similar method we find that. Therefore, the mixed strategy equilibrium is: Nina holds up one finger with 53.8% probability and two fingers up with 46.2% probability; Margaret holds up one finger and two fingers up with 50% probability.
3. Using backward induction:
Start by looking at player 1's decision in the final round. 1's best decision is to take the money. Using this information, we now focus on player 2's decision in the second round knowing what 1 will do in round 3. Player 2's best choice is to take the money and not wait.
Finally, we can now focus on player 1's decision in round 1, knowing how 2 will react to if 1 waits. Therefore, the Nash equilibrium for this game is that player 1 takes the $1 in round one and ends the game.
Allowing for additional rounds would not change the outcome of the game because backward induction would always lead us to the same outcome. The strategy of "wait" is strictly dominated by the strategy of "take."
5. There are two Nash equilibria: (80, 375) and (375, 80).
FIRM A
FIRM B
A possible grim strategy that could be devised is one in which both firms agree to charge high, and if one firm decides to price low the other firm will punish them permanently by pricing low from that period on. This would only work in a multi- or infinite-stage game, otherwise the threat of punishment would not be credible.
Chapter 13
. Experiments have allowed economist to study things such as fairness, status and trust; issues that standard economic theories did not account for. Experiments have also allowed economists to gather information on a wide range of contexts that are difficult to find in real world data.
3. Stating that $150 as the dollar amount that is just as good as the gamble when you took the $100 with violates the ranking principle and represents a choice reversal: two staples of standard economic theory. There is evidence that reversals decrease when decisions are repeated so that people learn from the consequences of their choices.
5. The default effect refers to the fact that when given many alternative choices, people often avoid making a choice and settle for the assigned default option. According to the default effect, the professor should have the in-class exam on the syllabus, but later offer the take-home exam and term paper as alternative options to the in-class exam. Evidence from economic experiments has shown that if the professor employs this strategy he/she should achieve the desired result.
7. Jason would be more likely to take advantage of the sale in the second scenario because, according to the concept of narrow framing, Jason would view each scenario as such: in (1) the TV is on sale, but it is less than 5% off, where in (2), the DVD player is 50% off, making this seem like a better deal, even though both scenarios offer a savings of $150.
9. First start by looking at Brian's utility from Sunday's perspective. Brian's utility will beif he washes Vincent on Monday,if he does it on Tuesday,if he does it on Wednesday, if he does it on Thursday, andif he does it on Friday.
To determine what Brian will actually do, analyze a few hypothetical situations. Suppose Wednesday arrives and Vincent is still dirty. Brian can either do it on Wednesday or wait until Thursday to wash Vincent. If he does it on Wednesday, his utility (from Wednesday's perspective) will be. Should he wait until Thursday from Wednesday's perspective his utility will be. Since 1 > -2, he will not wash Vincent on Thursday.
Now, suppose Tuesday rolls around and Vincent is still dirty. Brian can either do it on Tuesday or delay. If he does it on Tuesday, his utility (from Tuesday's perspective) will be. If he delays he will end up washing Vincent on Wednesday, so his utility (from Tuesday's perspective) will be 10 - 6 = 4. Since 4 > 1, Brian will do the chore on Wednesday.
On Monday, Brian can either do the chore or delay. Should he wash Vincent on Monday his utility from Monday's perspective will be. If he postpones, Vincent won't be clean until Wednesday, so his utility (from Monday's perspective) will be 10 - 6 =4. Since 4 = 4, he is indifferent between washing Vincent on Monday or Wednesday.
As of Sunday Brian will want to wash Vincent on Monday. Based on the reasoning above, Brian will wash Vincent on Monday. He is dynamically consistent, and he will follow through on Sunday's intentions.
1. First start by looking at Brian's utility from Sunday's perspective. Brian's utility will beif he washes Vincent on Monday,if he does it on Tuesday,if he does it on Wednesday, if he does it on Thursday, andif he does it on Friday. So as of Sunday, Monday is the best option.
Suppose Wednesday arrives and the litter box still stinks. Brian can either do it on Wednesday or wait until Thursday to clean the litter box. If he does it on Wednesday, his utility (from Wednesday's perspective) will be. Should he wait until Thursday from Wednesday's perspective his utility will be. Since 6 > 2, he will clean the litter box on Wednesday.
Now, suppose Tuesday rolls around and the litter box still stinks. Brian can either do it on Tuesday or delay. If he does it on Tuesday, his utility (from Tuesday's perspective) will be. If he delays he will end up cleaning the litter box on Wednesday, so his utility (from Tuesday's perspective) will be. Since 10 > 6, Brian will do the chore on Tuesday, because delaying would decrease his utility.
On Monday, Brian can either do the chore or delay. Should he clean the litter box on Monday his utility from Monday's perspective will be. If he postpones, the litter box will stink until Tuesday, so his utility (from Monday's perspective) will be. Since 16 > 6, he will clean the litter box Monday.
As of Sunday Brian want to clean the litter box on Monday. Based on the reasoning above, Brian will clean the litter box on Monday. He is dynamically consistent, and he will follow through on Sunday's intentions.
3. If Jason believes in the gambler's fallacy he will bet on black because he will reason that red is less likely. If Jason believes in the hot-handed fallacy he will bet on red because he will think that red is more likely to come up because it keeps coming up. The reasoning for both bets is in error, because the probability of red or black is equally likely and independent of prior outcomes.
5. We need to solve for the expected valuation of each risky prospect using expressions for positive and negative values of X.
If X = 10, then. The weight attached to is , while the weight attached to is.The value of this option is .
If X = 100, then. The weight attached to is , while the weight attached to is .The value of this option is
Jason would accept the first gamble that pays $10. He would not accept the $100 gamble because it offers negative payoffs.
Chapter 14
. The three main characteristics of a perfectly competitive market are: buyers and sellers don't have transaction costs; products are homogenous meaning that every seller produces the same product and consumers can't tell what seller produced what; and lastly, there are many sellers in the market such that every seller has a fraction of the market supply.
The presence of these factors inhibits sellers from charging a price above the market price. The large number of sellers in the market coupled with their lack of market power means that each firm must price at the market price; any firm that prices above the market price will lose their customer's. The fact that goods are homogenous means that if one firm decides to price above the market price customer's will not miss this firms product because every other firm in the market offers the same good. Thus, firms in a perfectly competitive market are price takers.
3.
To find the market demand, first find the respective inverse demand functions and solve for the P-intercept:
; P-intercept = 5.
; P-intercept = 6.
Now we may proceed as follows:
At prices below $5 only the market demand is. For prices between $5 and $6 dollars, only Jerry buys pizza, so the market demand function is. At prices above $6, neither Jerry nor, Marie demand pizza.
5. To determine the long-run market supply curve first find the output level at which marginal cost equals average cost.
Substituting this quantity into the expression for average cost tells us that
is the price at which the long-run market supply curve is a horizontal line. At prices under $4.12, the long-run supply is zero; and at prices above $4.12, it is infinite.
7. The long-run market equilibrium price equals the; we found the price to be $4.12, in problem (5). Now we need to find out how many pizzas are produced and sold every day in equilibrium. To find this we must evaluate the total quantity demanded at $4.12:. Lastly, in a long-run equilibrium each active firm produces 42.43 pizzas. This means that there are about 5.61 firms.
9. No, the long-run price will not always remain the same. If the demand in a market doubles, the price would almost certainly increase; however, in the long-run in a competitive and free entry market the increase in demand would attract new producers that would dissolve the short-run producer's surplus, driving the price back-down to the competitive level.
1. In this instance the firm's avoidable cost of making 180 slices can be found by measuring the area under the market supply curve up to the quantity of 180. If Q = 180, then. The supply curve may be rewritten as.
The avoidable cost of making 220 slices is,
3. First, solve for both inverse the demand and inverse supply functions. and, . Equate the function for price in terms of quantity demand to the function for price in terms of quantity supply to solve for the market equilibrium.
If we were to draw a horizontal line from, consumer surplus would be the area above this line but below the curve, thus
If we were to take the same horizontal line from, the producer surplus would be the area below this line but above the curve, thus
Aggregate surplus represents a total surplus which includes both CS and PS. Thus aggregate surplus, AS = 7500.
5. First, solve for both inverse the demand and inverse supply functions. and, . Equate the function for price in terms of quantity demand to the function for price in terms of quantity supply to solve for the market equilibrium.
If we were to draw a horizontal line from, consumer surplus would be the area above this line but below the curve, thus
If we were to take the same horizontal line from, the producer surplus would be the area below this line but above the curve, thus
Aggregate surplus represents a total surplus which includes both CS and PS. Thus aggregate surplus,.
Chapter 15
. An 18 cent tax on each gallon of gas is a specific tax. A 7.56% payroll tax is and ad valorem tax. An 8.25% sales tax is also an ad valorem tax. A $1.25 tax on cigarettes is a specific tax.
3. To find the competitive market equilibrium set the market demand function equal to the market supply function and find the equilibrium price, then use the price to find the equilibrium quantity.
If P = $100, then.
Now we need to find the consumer surplus, producer surplus and aggregate surplus.
The $15 tax on each down blanket changes the market supply function to
where Pb is the price paid by consumers. Now we need to find the equilibrium level of Pb, and again we set demand equal to supply:
If Pb = $110, then sellers must receive Ps = $95 per down blanket. Substituting Pb = $110 we find that blankets are sold.
Next, we need to find the consumer surplus, producer surplus, aggregate surplus, and now the deadweight loss.
To find the deadweight loss we need to look at the difference in quantities pre- and post-tax and the difference in prices between consumers and producers post-tax.
The deadweight loss of this tax is $75.
5. To find the competitive market equilibrium set the market demand function equal to the market supply function and find the equilibrium price, then use the price to find the equilibrium quantity.
If P = $1.5, then.
Now we need to find the consumer surplus, producer surplus and aggregate surplus.
The $0.50 tax on each refrigerator magnet changes the market supply function to
where Pb is the price paid by consumers.
Now we need to find the equilibrium level of Pb, and again we set demand equal to supply:
If Pb = $1.85, then sellers must receive Ps = $1 per refrigerator magnet. Substituting Pb = $1.85 we find that refrigerator magnets are sold.
Next, we need to find the consumer surplus, producer surplus, aggregate surplus, and now the deadweight loss.
To find the deadweight loss we need to look at the difference in quantities pre- and post-tax and the difference in prices between consumers and producers post-tax.
The deadweight loss of this tax is 0.18.
7. From, question (4) we know that the competitive equilibrium before the subsidy is P = $2 and Q = 18.
From question (4) we also know that consumer surplus, producer surplus and aggregate surplus are:
The $0.18 subsidy on each slice of pizza changes the market supply function to
where Pb is the price paid by consumers. Now we need to find the equilibrium level of Pb, and again we set demand equal to supply:
If Pb = $1.88, then sellers must receive Ps =$2.18 per pizza slice. Substituting Pb = $1.88 we find that slices of pizza are sold.
Next, we need to find the consumer surplus, producer surplus, aggregate surplus, and now the deadweight loss.
To find the deadweight loss we need to look at the difference in quantities pre- and post-subsidy and the difference in prices between consumers and producers post-subsidy.
The deadweight loss of this pizza subsidy is 0.18.
9. As we found in question (3) the equilibrium without interventions has a price of $100 per down blanket and 100 blankets are bought and sold each day. At a price of $120, consumers demand 80 blankets while blanket producers will want to supply 140 down blankets. The price floor policy would state that a down blanket could not be sold for more than $120. The price support program would have the government purchase 60 blankets each year, which is the difference between the quantity demanded and quantity supplied at $120. A quota would distribute permits to down blanket producers for selling 80 down blankets. In a voluntary production reduction program, the government would pay down blanket producers to reduce their production from 140 blankets to 80 blankets per day.
Now we must calculate the deadweight loss for each type of intervention: (use Figure 15.13 on p.558 of your textbook)
Price floor:
($90 is the price suppliers would charge if consumers would demand only 80 blankets)
Price support program:
A quota program:
The voluntary reduction production program:
1. A price ceiling at $85 for down blankets would restrict down blankets from being sold at a price below $85. This would reduce the number of blankets produced by suppliers and the amount demanded. Since supply is less than demand, the amount supplied would determine the new equilibrium quantity
3. The impact on social welfare would be in the form of a deadweight loss to consumers. Which may be calculated as follows (use Fig.15.18, on p.567 of the textbook):
5. The impact on social welfare would be in the form of a deadweight loss to consumers. Which may be calculated as follows (use Fig.15.18, on p.567 of the textbook):
Chapter 16
. The difference between partial equilibrium analysis and general equilibrium analysis is that general equilibrium analysis focuses on the competitive equilibrium in many markets at the same time, as opposed to, partial equilibrium analysis which only considers the equilibrium in a single isolated market.
The advantage of using general equilibrium analysis is that it allows economists to understand the consequences of interdependence in markets. Through the use of general equilibrium analysis economists can observe feedback between markets that have an effect on each other (i.e. market for crude oil and the market for automobiles).
3. First, notice that ice cream and cake are substitutes because the market-clearing curve is upward sloping. So, an increase in the price of ice cream would increase the demand for cake, which in turn increases the partial equilibrium price of cake.
5. First we must find the market clearing formulas for both deli turkey and roast beef. This is done by equating supply and demand in each market.
Deli Turkey:
Roast Beef:
Next, we must solve for the prices that satisfy both market-clearing formulas at the same time. This is done by substituting one into the other, like so:
The price of roast beef is therefore,
Plugging these values into either the supply or demand function, we find that in the general equilibrium, consumers buy 16 million pounds of deli turkey meat, and 42.8 million pounds of roast beef.
If the government imposes a 48 cent tax per pound of turkey meat, this changes the supply function for deli turkey meat to:
Once again equate demand and supply in the deli turkey market to find:
We can now proceed as before, substituting PR into PT:
The price for roast beef is thus:
Plugging these values into either the supply or demand function, we find that in the general equilibrium, consumers buy 13.6 million pounds of deli turkey meat, and 42.16 million pounds of roast beef.
7. The northeast boundary of this area (in other words the curve above) represents the utility possibilities frontier, which contains all the Pareto efficient allocations. Therefore, we can say that point A, B, and D are all Pareto efficient, but point C is not. It Marie and Jerry are starting off at point C then, we can make both better off by moving to point B.
9. The main difference between utilitarianism and Rawlsianism is that utilitarianism says that society should place equal weight on the well-being of every individual. Whereas, Rawlsianism says that as a society we should place all weight on the well-being of its worst-off member.
The difficulty in implementing either outcome-oriented principle is that they both require that we know a person's cardinal utility; a feat that most economists agree is an unachievable objective. Standard utility theory is based on the fact that we may only know consumer's ordinal preferences (which bundle they prefer when compared to another bundle).
1. The contract curve of an Edgeworth box identifies every efficient allocation of consumption goods. This turns out to be very useful because the contract curve shows every Pareto efficient outcome of consumption goods.
3. Marie's utility function is characteristic of goods that are deemed substitutable, thus Marie's food consumption does not have to equal her water consumption, but if the price of one good should increase in comparison to the price of the other good, she will consume less of that good and substitute it for the other good.
5. The laissez-faire approach to efficiency proposes that government should not interfere with markets in the form of regulations, because markets are perfectly capable of correcting themselves. Any form of government intervention in markets would produce inefficiencies.
The laissez-faire approach typically fails when there are market failures. Market failures are a source of inefficiency. Furthermore, real economies are hardly if ever perfectly competitive. The laissez-faire approach may also fail because it can produce inequitable results.
Chapter 17
. A firm that possesses market power is a firm that can profitably charge a price above its marginal cost. In a monopoly market there is only one seller, thus this monopolist firm would have possession of this market's power.
3. First find the inverse demand function, which is, so the. Using the formula, Boiler Pizza's marginal revenue when it sells Q slices of pizza is:
So Boiler Pizza's marginal revenue when it sells 5 slices is $3.6. This is less than its price at 5 slices of pizza, which is.
5. In problem (3) we found that Boiler Pizza's inverse demand function is, and its marginal revenue is . We can find its profit maximizing sales quantity and the associated price by applying the quantity rule and equating MR = MC:
Now, we can substitute this quantity into the inverse demand function. Thus, we find the price associated with the profit maximizing quantity is
We still need to check the shut-down rule. Observe that Boiler Pizza's profit is. Since Boiler Pizza earns nothing by shutting down, it will choose to operate at its profit-maximizing price of $3.80.
7. The Lerner Index says that the markup is equal to. To use this index we need to find the MC (500), which is 7.5. Therefore the mark-up is:
To find the elasticity of demand, use the fact that. Thus, in our example is.
9. To calculate the deadweight loss we need to first calculate the competitive equilibrium price and quantity. This can be found by setting equating marginal cost and the demand function for pizza.
=
Thus, price in the competitive equilibrium is:
From question (5) we retrieve that the monopolistic profit-maximizing equilibrium is Q = 9 and P = $3.80. We are now set to find the deadweight loss (Note: this is a simple matter in this example because the MC curve and demand curve are linear functions).
1. Rent seeking refers to the social waste that is the effort made in securing a monopoly. Rent seeking could result in losses beyond the standard deadweight loss of having a monopoly market if the monopoly had to use their anticipated monopoly profit to secure the monopoly.
3. To find Marty Mart's marginal expenditure, ME, first find the inverse supply function, and then evaluate how much they would have to pay to buy 2000 toaster ovens.
Thus, to buy 2000 toaster ovens Marty Mart would pay:
Sincefor this inverse supply function, Marty Mart's marginal expenditure is
So Marty Mart's marginal expenditure when it buys 2000 toaster ovens is $280, which is $200 more than it must pay to buy the 2000 toaster ovens.
5. To find the profit-maximizing number of toasters we need to derive the marginal benefit, which is given by the value of the inverse demand for toaster ovens. Since the demand curve is, then the inverse demand is. Now, we can proceed and find the profit-maximizing number of toaster ovens by setting MB = ME (from problem #13 we know):
=
The price to buy 1066.67 toaster ovens can be found by using the inverse demand function:
The price and quantity in a competitive equilibrium could be found by equating supply and demand:
=
Chapter 18
. Perfect price discrimination is what can occur if a monopolist knows each consumer's willingness to pay for each unit he sells, and is thus able to charge different prices for each unit. This is just about impossible to do because of the sheer amount of access to personal information on each consumer a firm would need. Firms are always trying to obtain more about their consumers, but perfect price discrimination is more of a theoretical possibility rather than an actual possibility.
3. A firm would like consumers with the low elasticity of demand to pay a higher price because they are more likely to accept increase in prices, because of their inelastic demand. This is like the example of the business traveler who has to travel on a moments notice and can't afford to look for a bargain.
5. To answer this question we first must solve for the quantity demand of each consumer at the competitive equilibrium (P = MC). Each consumer will demand. To calculate consumer surplus we need the inverse demand function. Consumer surplus is $175.09 which can be calculated taking a height of 5.33 minus 0.5 and multiplying by the base 72.5 times one-half (area of a triangle).
The consumer surplus represents the maximum amount that a monopolist firm could set a two-part tariff at. Any amount past $175.09 will cause the consumer to demand less quantity. Also, at a tariff of $175.09 Dolly will not be able to charge for a subsequent fills of there special tub. Therefore, Dolly's profit will be 200 times the amount of consumer surplus or the two-part tariff, $35,018.
7. We found in the previous problem that price discrimination hurts multiple-cat owners by increasing the price, but single-cat owners are helped by a lowering of the price.
The gain for single-cat owners is equal to the area of the trapezoid with height 3.80 minus 3.50, and one base equal to 2.4, while the other base is equal to 3 minus 2.4. This yields a consumer surplus of,
The loss in consumer surplus to multiple-cat owners is equal to the area of the trapezoid with height 4 minus 3.8, and one base equal to 2.4, and the other base equal to 6.6 minus 6. This yields a consumer surplus of,
The total net loss on consumer surplus due to the use of price discrimination is. The gain in profit, as we found in the previous problem, was 0.3. So we can conclude that price discrimination lowers aggregate surplus by 0.15 (.
9. We found in the previous problem that price discrimination hurts consumers on hot days because it increases the price, but consumers on cold days are helped by a lowering of the price. The gain for cold day consumers is equal to the area of the trapezoid with height 0.626 subtract 0.501, and one base equal to 1.987, while the other base is equal to 1.989 subtract 1.987 (note: to get the numbers for the base we simply plug-in the prices ex-post and ex-ante of price discrimination with concerns to cold day consumers). This yields a consumer surplus of,
The loss in consumer surplus consumers on hot day is equal to the area of the trapezoid with height 0.75 minus 0.626, and one base equal to 2.985, and the other base equal to 2.987 minus 2.985. This yields a loss of consumer surplus of,
The total net loss on consumer surplus due to the use of price discrimination is. The gain in profit, as we found in the previous problem, was 0.065. So we can conclude that price discrimination lowers aggregate surplus by, .
1. The table below shows four different per pound prices, the quantities purchased by each type of consumer, the fixed fee, the profits form sales to each type of consumer, and the total profit. To calculate the quantity at a certain price just plug in the price into the corresponding demand function that was given. The fixed fee is the same as the quantity of single-cat owner consumer surplus. Profits per type of consumer are calculated by multiplying the specific price and quantity, then adding the fixed fee and finally subtracting the marginal cost times the number of pounds purchased. To find total profit, take the profit for each type of consumer and aggregate it across the whole market (i.e. all single-cat owners and multiple-cat owner).
According to the chart below the most profitable price for Dolly to charge would be $2.00 earning them a profit of $3,900.
Price per pound
$1.00
$1.50
$2.00
$2.50
Pounds purchased
Single-cat owner
8
7
6
5
Multiple-cat owner
5
3.5
2
0.5
Fixed fee ($):
$16
$12.25
$9
$6.25
Profits ($):
Single-cat owner
$8
$8.75
$15
$8.75
Multiple-cat owner
$16
$5.5
$9
$11.50
Total profit ($):
$3,200
$2,300
$3,900
$2,900
3. The table below shows four different per pound prices, the quantities purchased by each type of consumer, the fixed fee, the profits form sales to each type of consumer, and the total profit.
To calculate the quantity at a certain price just plug in the price into the corresponding demand function that was given. The fixed fee is the same as the quantity of single-cat owner consumer surplus. Profits per type of consumer are calculated by multiplying the specific price and quantity, then adding the fixed fee and finally subtracting the marginal cost times the number of pounds purchased. To find total profit, take the profit for each type of consumer and aggregate it across the whole market (i.e. all single-cat owners and multiple-cat owner).
According to the chart below, the most profitable price for Dolly to charge would be $2.00 earning them a profit of $5,700.
Price per pound
$1.00
$1.50
$2.00
$2.50
Pounds purchased
Single-cat owner
8
7
6
5
Multiple-cat owner
5
3.5
2
0.5
Fixed fee ($):
$16
$12.25
$9
$6.25
Profits ($):
Single-cat owner
$8
$8.75
$15
$8.75
Multiple-cat owner
$16
$5.5
$9
$11.50
Total profit ($):
$6,400
$3,400
$5,700
$5,200
5. Bundling is the practice of selling several products together as package. Firms can use bundling to extract additional consumer surplus because different consumers' willingness to pay for products that are negatively related. A monopolist will use bundling to eliminate the variation in consumer valuations, allowing the monopolist to extract the entire aggregate surplus as profit.
Chapter 19
. The Nash equilibrium in a Bertrand oligopoly model is at P = MC. At any price above MC a firm won't sell any output because they will be under-cut by other another firm(s) that will price at MC. At any price below MC the firm will sell its output but will lose money because P > MC. All that is needed in a single market for this to happen is two firms.
3. In a Bertrand model the equilibrium price equal to marginal cost. In this case it is $20. To find the equilibrium quantity set P = MC.
Thus, 300 handbags will be bought and sold at a price of $20 in this market. Since, both Kristen and Megan are pricing the handbags at marginal cost they will earn $0 in economic profit. To see this numerically calculate revenue, which is the price, multiplied by the quantity sold, which is $6000 in this example. Now, we have to subtract marginal cost,$20, times the number of units sold 300. Thus profit is $6000 - $6000 = 0.
5. In order to solve for the Nash equilibrium output there are several things which we must do first. The market inverse demand is. Given Kristin and Megan's outputs we find that the price is. We can rewrite this expression in terms of residual demand as follows:
Kristin:
, and similarly
Megan:
.
Now we need to find Kristin and Megan's best-response function which entails that we find their respective marginal revenue functions as well as their profit maximizing output. Remember that.
Kristin:
Kristin's profit maximizing output can be found by setting MR = MC.
Because of the symmetrical nature of the problem we can deduce without calculation that Megan's best-response function is. (You can make sure this is so, as an exercise).
To find the Nash equilibrium quantity we substitute Kristin's best-response function into Megan's best-response function, or vice versa.
Substitute this quantity into Kristin's best-response function and we find that, which gives us the Nash equilibrium quantity.
Substitute the equilibrium quantity into the inverse market demand function which is the equilibrium price (notice that it is $30 above the marginal cost). Thus both Megan and Kristin each make a profit of $13.33 times 66.7 (quantity), which is $890.04.
7. There are a few steps we must do before we solve for the Nash equilibrium. First should rearrange the demand functions in a form that will better suit our analysis. The functions for Joe's and Bob's quantity demanded given above can be rewritten as follows.
Given this form above we can solve for both Joe's and Bob's inverse demand functions. They are
Now we proceed by finding Joe's best response function, which will involve deriving his marginal revenue (like we have done before), where ; and solving for the profit maximizing quantity.
Solving for the profit maximizing quantity requires that we set MR = MC.
Rearranging terms will yield the profit-maximizing sales quantity
Next, we should solve for the profit-maximizing sales price. To accomplish this we could simply substitute our profit-maximizing quantity into Joe's inverse demand function.
Again, we may use the symmetrical nature of the problem to conclude that Bob's best response function is. If you feel that you need more practice you could derive Bob's profit-maximizing sales price in the same way we found Joe's.
Now, we can find the Nash equilibrium by substituting Joe's profit-maximizing sales price into Bob's profit-maximizing sales price, or vice a versa.
Similarly, the Nash equilibrium price for Joe's is also $1.83 per hamburger.
To find what quantities will be sold by each stand we can substitute in our Nash equilibrium prices into our original demand function.
To find how much profit will be made by each stand take the difference between the equilibrium price and the marginal cost (0.83), and then multiply by the equilibrium quantity (104.25) which will yield each stand a profit of about $86.53.
9. Kristen will not undercut Megan as long as the following holds:
Equivalently, the inequality above can be expressed by. The 1 indicates an interest rate of 100%, meaning that Kristen will not undercut Megan unless the interest rate was 100% or higher.
1. The difference between tacit collusion and explicit collusion is that in the former case colluding firms communicate with each other and have understood agreements, whereas tacit collusion colluding firms do not communicate but an understanding is reached implicitly.
Tacit collusion between firms is harder to maintain because they are made without communication, which makes a mutual understanding between colluding firms very difficult to achieve.
3. Recall Kristin's inverse residual demand function from question 4.
Also, recall Megan's best response function from question 4.
If we substitute Megan's best response function into Kristin's inverse residual demand function, the result will be a formula that describes the price Kristin's receives at each level of output, while taking into account what Megan's best response will be to her choice:
From this relationship, Kristin can derive her marginal revenue, which is:
Kristin will maximize her profit by choosing the output levelequates her marginal revenue with her marginal cost:
Megan's best response to this output is. At a total output of 225 handbags the market price is $27.50. Kristin's profit is $1,125 which is $875 less than what she made if the two chose their inputs simultaneously, while Megan profits $562.50 ($1437.50 less than in the Cournot example).
5. A horizontal merger is occurs when two or more competing firms merge (that is combine) their operations, and become one firm. A horizontal merger would be more effective than fixing prices because fixing prices involves an agreement between two firms that must be enforced in order to properly fix a price. A horizontal merger between the firms that were looking to fix prices would simplify the desired outcome greatly.
Chapter 20
. When an agent(s) action(s) create an externality it affects someone the agent has not engaged in a related market transaction. A negative externality is simply an externality that harms someone else.
Negative externalities tend to create deadweight losses, particularly in competitive markets because competitive markets are usually unable allocate resources efficiently under such circumstances.
3. Since we know the MEC curve and the MC in the market (given by the supply curve), we can deduce from the formula MSC = MEC + MC, that The shaded region represents the deadweight-loss.
5. Let us first find the competitive equilibrium. To do so, rewrite the marginal cost curve solving for P (solve for inverse demand of a single firm). Remember in competitive equilibrium we set P = MC. We get that, which we now have to have to multiply by 10 (10 firms in the market) to the market supply curve, . Now we can set supply equal to demand and find competitive equilibrium quantities.
Plugging in the price into either the demand or supply function shows that, Q = 75.
Now we must solve for the efficient quantity. To come to an efficient quantity we know that the construction firms will be responsible for external costs of their production. Thus, P = MSC = MC + MEC =. Solving for quantity in terms of price we get. Multiplying this previous result by 10, we get the market supply curve,. Now equate demand and supply.
Substituting the price into the demand function we get that the efficient quantity level is Q =-4.5, so it is best not to produce any more construction sites Q = 0.
The deadweight loss created the firms is shown is similar to that shown on figure 20.1 in your textbook (p.756). Take the 75 construction sites under competitive equilibrium and substitute it in for Q in the MEC function, which yields 160. The height of the triangle is the difference between the 75 sites in competitive equilibrium and the 0 efficient quantity. The deadweight loss is found by computing the area of the triangle.
7. An emission standard is a legal limit on the amount of pollution that a person or entity can produce from a particular activity. In order for the government to set an efficient emission standard it needs to know precisely the polluter's cost of abatement and the social costs. Thus, both sides have an incentive to overestimate the costs. On the one hand, the polluter will overestimate their cost of abatement so that the government places higher emission standards. The pollutee has the incentive to overestimate the social cost of emissions so that the government sets a lower emission standard.
9. The efficient number of gnomes is found by equating the marginal benefit to the marginal cost.
At the efficient level of gnomes your marginal cost is $8. Therefore, an efficient Pigouvian tax is $8 per gnome. With such a tax your neighbor will choose to only 8 gnomes and will pay a total of $48 dollars to have the gnomes running free in the garden.
1. A liability rule is a legal principle that forces polluters that create an externality to compensate the affected parties for some or all of their losses. So a liability rule would make a polluter internalize the marginal cost of the externality they create. One of the difficulties that arise with implementing a liability rule are legal fees which are costly and time consuming.
3. First, we must compute the MSB of having plants in the classroom.
Now we set MSB = MC:
Thus the socially efficient number of plants is 16. To find what each student would chose individually set MB =MC:
Individually each student would choose to have 12 plants. If you notice, at Q = 16 the marginal cost exceeds the marginal benefit, such that each student will have an incentive to reduce the number of plants. The break-even point between marginal cost and marginal benefit is where Q = 12.
5. The median voter theorem is refers to voters with single-peaked preferences, in which a majority of them prefer the median ideal policy to all other policies. This theorem requires that the policy of the median voter coincide with the social benefit policy. In reality, they rarely coincide.
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