16 BUDGET CONSTRAINT (CI Harold contemplates the set of commodity bundles that it could afford by making one declaration or the other. Let us call a commodity bundle "attainable"if Harold can afford it by declaring itself to be a "Blue and buying the bundle with blue money or if Harold can afford the bundle by declaring itself to be a"Red"and buying it with red money. On the diagram below, shade in all of the attainable bundles 15 Blue line 5 ErRed Line Ambrosia 2. 13(0) Are Mungoan budgets really so fanciful? Can you think of sit- uations on earth where people must simultaneously satisfy more than one budget constraint? Is money the only scarce resource that people use up when consuming? Consumption of many commodities takes time as well as money. People have to simultaneously satisfy a time budget nd a money budget. other examples--people may have a calorie budget or a cholesterol budget or an alcohol-intake budget
16 BUDGET CONSTRAINT (Ch. 2) Harold contemplates the set of commodity bundles that it could afford by making one declaration or the other. Let us call a commodity bundle “attainable” if Harold can afford it by declaring itself to be a “Blue” and buying the bundle with blue money or if Harold can afford the bundle by declaring itself to be a “Red” and buying it with red money. On the diagram below, shade in all of the attainable bundles. 0 5 10 15 20 5 10 15 Ambrosia Gum 20 Blue Line Red Line 2.13 (0) Are Mungoan budgets really so fanciful? Can you think of situations on earth where people must simultaneously satisfy more than one budget constraint? Is money the only scarce resource that people use up when consuming? Consumption of many commodities takes time as well as money. People have to simultaneously satisfy a time budget and a money budget. Other examples--people may have a calorie budget or a cholesterol budget or an alcohol-intake budget
Chapter 3 NAM Preferences Introduction. In the previous section you learned how to use graphs to show the set of commodity bundles that a consumer can section, you learn to put information about the consumer's preferences the same kind of graph. Most of the problems ask you to draw indifference curves Sometimes we give you a formula for the indifference curve. Then all you have to do is graph a known equation. But in some problems, we give you only "qualitative"information about the consumer's preferences and ask you to sketch indifference curves that are consistent with this information. This requires a little more thought. Don't be surprised or disappointed if you cannot immediately see the answer when you look at a problem, and don't expect that you will find the answers hiding somewhere in your textbook. The best wS aLCE swer and label them, re know to find answers is to “ think and doodle.” Draw some axes on sc then mark a point on your graph and ask yourself, " What other points on the graph would the consumer find indifferent to this point? If possible. draw a curve connecting such points, making sure that the shape of the line you draw reflects the features required by the problem. This gives you one indifference curve. Now pick another point that is preferred to the first one you drew and draw an indifference curve through it EXample: Jocasta loves to dance and hates housecleaning. She has strictly convex preferences. She prefers dancing to any other activity and never gets tired of dancing, but the more time she spends cleaning house, the less happy she is. Let us try to draw an indifference curve that is consistent with her preferences. There is not enough information here to tell us exactly where her indifference curves go, but there is enough information to determine some things about their shape. Take a piece of scratch paper and draw a pair of axes. Label the horizontal axis"Hours per day of housecleaning. "Label the vertical axis"Hours per day of dancing. " Mark a point a little ways up the vertical axis and write a 4 next to it. At this point, she spends 4 hours a day dancing and no time housecleaning. Other points that would be indifferent to this point would have to be points where she did more dancing and more housecleaning. The pain of the extra housekeeping should just compensate for the pleasure of the extra dancing. So an indifference curve for Jocasta must be upward sloping Because she loves dancing and hates housecleaning, it must be that she prefers all the points above this indifference curve to all of the points on or below it. If Jocasta has strictly convex preferences, then it must be that if you draw a line between any two points on the same indifference curve, all the points on the line(except the endpoints) are preferred to the endpoints. For this to be the case, it must be that the indifference curve slopes upward ever more steeply as you move to the right along it You should convince yourself of this by making some drawings on scratch
Chapter 3 NAME Preferences Introduction. In the previous section you learned how to use graphs to show the set of commodity bundles that a consumer can afford. In this section, you learn to put information about the consumer’s preferences on the same kind of graph. Most of the problems ask you to draw indifference curves. Sometimes we give you a formula for the indifference curve. Then all you have to do is graph a known equation. But in some problems, we give you only “qualitative” information about the consumer’s preferences and ask you to sketch indifference curves that are consistent with this information. This requires a little more thought. Don’t be surprised or disappointed if you cannot immediately see the answer when you look at a problem, and don’t expect that you will find the answers hiding somewhere in your textbook. The best way we know to find answers is to “think and doodle.” Draw some axes on scratch paper and label them, then mark a point on your graph and ask yourself, “What other points on the graph would the consumer find indifferent to this point?” If possible, draw a curve connecting such points, making sure that the shape of the line you draw reflects the features required by the problem. This gives you one indifference curve. Now pick another point that is preferred to the first one you drew and draw an indifference curve through it. Example: Jocasta loves to dance and hates housecleaning. She has strictly convex preferences. She prefers dancing to any other activity and never gets tired of dancing, but the more time she spends cleaning house, the less happy she is. Let us try to draw an indifference curve that is consistent with her preferences. There is not enough information here to tell us exactly where her indifference curves go, but there is enough information to determine some things about their shape. Take a piece of scratch paper and draw a pair of axes. Label the horizontal axis “Hours per day of housecleaning.” Label the vertical axis “Hours per day of dancing.” Mark a point a little ways up the vertical axis and write a 4 next to it. At this point, she spends 4 hours a day dancing and no time housecleaning. Other points that would be indifferent to this point would have to be points where she did more dancing and more housecleaning. The pain of the extra housekeeping should just compensate for the pleasure of the extra dancing. So an indifference curve for Jocasta must be upward sloping. Because she loves dancing and hates housecleaning, it must be that she prefers all the points above this indifference curve to all of the points on or below it. If Jocasta has strictly convex preferences, then it must be that if you draw a line between any two points on the same indifference curve, all the points on the line (except the endpoints) are preferred to the endpoints. For this to be the case, it must be that the indifference curve slopes upward ever more steeply as you move to the right along it. You should convince yourself of this by making some drawings on scratch
18 PREFERENCEs (Ch. 3) paper. Draw an upward-sloping curve passing through the point(0, 4) and getting steeper as one moves to the right When you have completed this workout, we hope that you will be able to do the following: . Given the formula for an indifference curve. draw this curve and find its slope at any point on the curve. Determine whether a consumer prefers one bundle to another or is indifferent between them, given specific indifference curves Draw indifference curves for the special cases of perfect substitutes and perfect complement . draw indifference curves for someone who dislikes one or both com- modities Draw indifference curves for someone who likes goods up to a point but who can get“to Identify weakly preferred sets and determine whether these are con- vex sets and whether preferences are convex . Know what the marginal rate of substitution is and be able to deter- mine whether an indifference curve exhibits diminishing marginal rate of substitution. Determine whether a preference relation or any other relation be- tween pairs of things is transitive, whether it is reflexive, and whether it 3.1(0) Charlie likes both apples and bananas. He consumes nothing else The consumption bundle where Charlie consumes a bushels of apples per year and B bushels of bananas per year is written as(IA, IB).Last year, Charlie consumed 20 bushels of apples and 5 bushels of bananas. It appens that the set of consumption bundles(A, IB) such that Charlie is indifferent between(aA, B) and(20, 5) is the set of all bundles such that aB= 100/ A. The set of bundles(A, aB) such that Charlie is just indifferent between(A, B) and the bundle(10, 15) is the set of bundle such that B= 150/ (a) On the graph below, plot several points that lie on the indifference curve that passes through the point(20, 5), and sketch this curve, using blue ink. Do the same, using red ink, for the indifference curve passing through the point(10, 15) (b) Use pencil to shade in the set of commodity bundles that Charlie weakly prefers to the bundle(10, 15). Use blue ink to shade in the set of commodity bundles such that Charlie weakly prefers(20, 5) to these bundle
18 PREFERENCES (Ch. 3) paper. Draw an upward-sloping curve passing through the point (0, 4) and getting steeper as one moves to the right. When you have completed this workout, we hope that you will be able to do the following: • Given the formula for an indifference curve, draw this curve, and find its slope at any point on the curve. • Determine whether a consumer prefers one bundle to another or is indifferent between them, given specific indifference curves. • Draw indifference curves for the special cases of perfect substitutes and perfect complements. • Draw indifference curves for someone who dislikes one or both commodities. • Draw indifference curves for someone who likes goods up to a point but who can get “too much” of one or more goods. • Identify weakly preferred sets and determine whether these are convex sets and whether preferences are convex. • Know what the marginal rate of substitution is and be able to determine whether an indifference curve exhibits “diminishing marginal rate of substitution.” • Determine whether a preference relation or any other relation between pairs of things is transitive, whether it is reflexive, and whether it is complete. 3.1 (0) Charlie likes both apples and bananas. He consumes nothing else. The consumption bundle where Charlie consumes xA bushels of apples per year and xB bushels of bananas per year is written as (xA,xB). Last year, Charlie consumed 20 bushels of apples and 5 bushels of bananas. It happens that the set of consumption bundles (xA,xB) such that Charlie is indifferent between (xA,xB) and (20, 5) is the set of all bundles such that xB = 100/xA. The set of bundles (xA,xB) such that Charlie is just indifferent between (xA,xB) and the bundle (10, 15) is the set of bundles such that xB = 150/xA. (a) On the graph below, plot several points that lie on the indifference curve that passes through the point (20, 5), and sketch this curve, using blue ink. Do the same, using red ink, for the indifference curve passing through the point (10, 15). (b) Use pencil to shade in the set of commodity bundles that Charlie weakly prefers to the bundle (10, 15). Use blue ink to shade in the set of commodity bundles such that Charlie weakly prefers (20, 5) to these bundles
NAME Red Cul Pencil Shading 10 lue cury For each of the following statements about Charlie's preferences, write true”or“ false. c)(30,5)~(10,15).True. (d)(10,15)>(20,5).True. (e)(20,5)≥(10,10,.True f)(24,4)≥(11,9.1).Fa1se g)(1,14)>(2,49).True. (h)a set is convex if for any two points in the set, the line segment between them is also in the set. Is the set of bundles that Charlie weakl prefers to(20, 5) a convex se Yes (i) Is the set of bundles that Charlie considers inferior to(20, 5)a convex ? ne () The slope of Charlie's indifference curve through a point, (a, 2b), is known as his marginal rate of substitution at that point
NAME 19 0 10 20 30 40 10 20 30 Apples Bananas 40 Blue Curve Pencil Shading Red Curve Blue Shading For each of the following statements about Charlie’s preferences, write “true” or “false.” (c) (30, 5) ∼ (10, 15). True. (d) (10, 15) (20, 5). True. (e) (20, 5) (10, 10). True. (f) (24, 4) (11, 9.1). False. (g) (11, 14) (2, 49). True. (h) A set is convex if for any two points in the set, the line segment between them is also in the set. Is the set of bundles that Charlie weakly prefers to (20, 5) a convex set? Yes. (i) Is the set of bundles that Charlie considers inferior to (20, 5) a convex set? No. (j) The slope of Charlie’s indifference curve through a point, (xA,xB), is known as his marginal rate of substitution at that point
0 PREFERENCEs ( Ch. 3) (k)Remember that Charlie's indifference curve through the point(10, 10) has the equation B= 100 /A. Those of you who know calculus will remember that the slope of a curve is just its derivative, which in this case is-100/A.(If you don't know calculus, you will have to take our word for this. Find Charlie's marginal rate of substitution at the point (What is his marginal rate of substitution at the point (5, 20)? -4 (m)What is his marginal rate of substitution at the point(20, 5)? 25 (n) Do the indifference curves you have drawn for Charlie exhibit dimin ishing marginal rate of substitution? Yes 3.2(0) Ambrose consumes only nuts and berries. Fortunately, he likes both goods. The consumption bundle where Ambrose consumes I units of nuts per week and 2 units of berries per week is written as(1, 12) The set of consumption bundles(1, T2)such that Ambrose is indifferent between(a1, a2)and(1, 16)is the set of bundles such that T120, 12>0, and 22=20-4var. The set of bundles(a1, r2)such that(a1,.r (36, 0)is the set of bundles such that 120, 2220 and 2=24-4VaI (a) On the graph below, plot several points that lie on the indifference curve that passes through the point(1, 16), and sketch this curve, using blue ink. Do the same, using red ink, for the indifference curve passing through the point(36, 0) (b) Use pencil to shade in the set of commodity bundles that Ambrose weakly prefers to the bundle(1, 16). Use red ink to shade in the set of all commodity bundles(a1, a2)such that Ambrose weakly prefers(36, 0) to these bundles. Is the set of bundles that Ambrose prefers to(1, 16)a es (c) What is the slope of Ambrose's indifference curve at the point( 9, 8)? (Hint: Recall from calculus the way to calculate the slope of a curve. If ou don't know calculus, you will have to draw your diagram carefull 3
20 PREFERENCES (Ch. 3) (k) Remember that Charlie’s indifference curve through the point (10, 10) has the equation xB = 100/xA. Those of you who know calculus will remember that the slope of a curve is just its derivative, which in this case is −100/x2 A. (If you don’t know calculus, you will have to take our word for this.) Find Charlie’s marginal rate of substitution at the point, (10, 10). −1. (l) What is his marginal rate of substitution at the point (5, 20)? −4. (m) What is his marginal rate of substitution at the point (20, 5)? (−.25). (n) Do the indifference curves you have drawn for Charlie exhibit diminishing marginal rate of substitution? Yes. 3.2 (0) Ambrose consumes only nuts and berries. Fortunately, he likes both goods. The consumption bundle where Ambrose consumes x1 units of nuts per week and x2 units of berries per week is written as (x1,x2). The set of consumption bundles (x1,x2) such that Ambrose is indifferent between (x1,x2) and (1, 16) is the set of bundles such that x1 ≥ 0, x2 ≥ 0, and x2 = 20 − 4 √x1. The set of bundles (x1,x2) such that (x1,x2) ∼ (36, 0) is the set of bundles such that x1 ≥ 0, x2 ≥ 0 and x2 = 24−4 √x1. (a) On the graph below, plot several points that lie on the indifference curve that passes through the point (1, 16), and sketch this curve, using blue ink. Do the same, using red ink, for the indifference curve passing through the point (36, 0). (b) Use pencil to shade in the set of commodity bundles that Ambrose weakly prefers to the bundle (1, 16). Use red ink to shade in the set of all commodity bundles (x1,x2) such that Ambrose weakly prefers (36, 0) to these bundles. Is the set of bundles that Ambrose prefers to (1, 16) a convex set? Yes. (c) What is the slope of Ambrose’s indifference curve at the point (9, 8)? (Hint: Recall from calculus the way to calculate the slope of a curve. If you don’t know calculus, you will have to draw your diagram carefully and estimate the slope.) −2/3