36 UTILITY(Ch. 4) 4.1(0)Remember Charlie from Chapter 3? Charlie consumes apples and bananas. We had a look at two of his indifference curves. In this problem we give you enough information so you can find all of Charlie's indifference curves. We do this by telling you that Charlie's utility function happens to be U(EA, 1B)=aATB (a) Charlie has 40 apples and 5 bananas. Charlie's utility for the bun- dle(40, 5)is U(40, 5)= 200. The indifference curve through(40, 5) includes all commodity bundles(aA, IB) such that zAIB= 200.So the indifference curve through(40, 5) has the equation B==. On the graph below, draw the indifference curve showing all of the bundles that Charlie likes exactly as well as the bundle(40, 5) Bananas 10 Appl (b)Donna offers to give Charlie 15 bananas if he will give her 25 apples Would Charlie have a bundle that he likes better than(40, 5) if he makes this trade? Yes. What is the largest number of apples that D could demand from Charlie in return for 15 bananas if she expects him to be willing to trade or at least indifferent about trading? 30.(Hint:If Donna gives Charlie 15 bananas, he will have a total of 20 bananas. If he has 20 bananas, how many apples does he need in order to be as well-off as he would be without trade? 4.2(0 Ambrose, whom you met in the last chapter, continues to thrive on nuts and berries. You saw two of his indifference curves. One indie- ference curve had the equation a2=20-4Val, and another indifference curve had the equation 2= 24-4Vai, where 1 is his consumption of
36 UTILITY (Ch. 4) 4.1 (0) Remember Charlie from Chapter 3? Charlie consumes apples and bananas. We had a look at two of his indifference curves. In this problem we give you enough information so you can find all of Charlie’s indifference curves. We do this by telling you that Charlie’s utility function happens to be U(xA,xB) = xAxB. (a) Charlie has 40 apples and 5 bananas. Charlie’s utility for the bundle (40, 5) is U(40, 5) = 200. The indifference curve through (40, 5) includes all commodity bundles (xA,xB) such that xAxB = 200. So the indifference curve through (40, 5) has the equation xB = 200 xA . On the graph below, draw the indifference curve showing all of the bundles that Charlie likes exactly as well as the bundle (40, 5). 0 10 20 30 40 10 20 30 Apples Bananas 40 (b) Donna offers to give Charlie 15 bananas if he will give her 25 apples. Would Charlie have a bundle that he likes better than (40, 5) if he makes this trade? Yes. What is the largest number of apples that Donna could demand from Charlie in return for 15 bananas if she expects him to be willing to trade or at least indifferent about trading? 30. (Hint: If Donna gives Charlie 15 bananas, he will have a total of 20 bananas. If he has 20 bananas, how many apples does he need in order to be as well-off as he would be without trade?) 4.2 (0) Ambrose, whom you met in the last chapter, continues to thrive on nuts and berries. You saw two of his indifference curves. One indifference curve had the equation x2 = 20 − 4 √x1, and another indifference curve had the equation x2 = 24 − 4 √x1, where x1 is his consumption of
nuts and a2 is his consumption of berries it can be told that am- brose has quasilinear utility. In fact, his preferences can be represented by the utility function U(1, 32)=4v1+I2 a)Ambrose originally consumed 9 units of nuts and 10 units of berries His consumption of nuts is reduced to 4 units, but he berries so that he is just as well-off as he was before. After the change how many units of berries does Ambrose consume? 14 (b)On the graph below, indicate Ambrose's original consumption and sketch an indifference curve passing through this point. As you can verify Ambrose is indifferent between the bundle(9, 10)and the bundle(25, 2) If you doubled the amount of each good in each bundle, you would have bundles(18, 20) and(50, 4). Are these two bundles on the same indiffer ence curve? No.(Hint: How do you check whether two bundles are indifferent when you know the utility function Berries 20 10 (9,10) (c) What is Ambrose's marginal rate of substitution, MRS(1, 2), when he is consuming the bundle(9, 10)?( Give a numerical answer. )-2/3 What is Ambrose's marginal rate of substitution when he is consumin the bundle(9, 20)?-2/3 (d)We can write a general expression for Ambrose's marginal rate of bstitution when he is cor commodity bundle(a1, 22). This is MRS(1,T2)=-2/VE1. Although we always write MRS(21, 22) s a function of the two variables, Ii and 2, we see that ambrose s utility function has the special property that his marginal rate of substitution does not change when the variable
NAME 37 nuts and x2 is his consumption of berries. Now it can be told that Ambrose has quasilinear utility. In fact, his preferences can be represented by the utility function U(x1,x2)=4√x1 + x2. (a) Ambrose originally consumed 9 units of nuts and 10 units of berries. His consumption of nuts is reduced to 4 units, but he is given enough berries so that he is just as well-off as he was before. After the change, how many units of berries does Ambrose consume? 14. (b) On the graph below, indicate Ambrose’s original consumption and sketch an indifference curve passing through this point. As you can verify, Ambrose is indifferent between the bundle (9,10) and the bundle (25,2). If you doubled the amount of each good in each bundle, you would have bundles (18,20) and (50,4). Are these two bundles on the same indifference curve? No. (Hint: How do you check whether two bundles are indifferent when you know the utility function?) 0 5 10 15 20 5 10 15 Nuts Berries 20 (9,10) (c) What is Ambrose’s marginal rate of substitution, MRS(x1,x2), when he is consuming the bundle (9, 10)? (Give a numerical answer.) −2/3. What is Ambrose’s marginal rate of substitution when he is consuming the bundle (9, 20)? −2/3. (d) We can write a general expression for Ambrose’s marginal rate of substitution when he is consuming commodity bundle (x1,x2). This is MRS(x1,x2) = −2/ √x1. Although we always write MRS(x1,x2) as a function of the two variables, x1 and x2, we see that Ambrose’s utility function has the special property that his marginal rate of substitution does not change when the variable x2 changes
38 UTILITY (Ch. 4) 4.3(0) Burt's utility function is U(a1, 2)=(a1+2)(a2 +6), where al is the number of cookies and 2 is the number of glasses of milk that he consumes (a) What is the slope of Burt's indifference curve at the point where he is consuming the bundle(4, 6)?-2. Use pencil or black ink to draw a line with this slope through the point (4, 6).(Try to make this graph fairly neat and precise, since details will matter. The line you just drew is the tangent line to the consumer's indifference curve at the point (4, 6) (b) The indifference curve through the point(4, 6) passes through the ),(7, 2 ) and(2, 12).Use blue ink ketch in this indifference curve. Incidentally, the equation for Burts difference curve through the point (4, 6)is r 72/(1 +2)-6 Glasses of milk b Blue curve ABlack Line Cookies (c)burt currently has the bundle(4, 6). Ernie offers to give Burt 9 glasses of milk if Burt will give Ernie 3 cookies. If Burt makes this trade, he would have the bundle(1, 15). Burt refuses to trade. Was this Mark the bundle(1, 15)on your graph 63< U(4, 6)=72 a wise decision? Yes, U(1, 15 (d) Ernie says to Burt, "Burt, your marginal rate of substitution is -2 That means that an extra cookie is worth only twice as much to you as n extra glass of milk. I offered to give you 3 glasses of milk for every cookie you give me. If I offer to give you more than your marginal rate of substitution, then you should want to trade with me. Burt replies
38 UTILITY (Ch. 4) 4.3 (0) Burt’s utility function is U(x1,x2)=(x1 + 2)(x2 + 6), where x1 is the number of cookies and x2 is the number of glasses of milk that he consumes. (a) What is the slope of Burt’s indifference curve at the point where he is consuming the bundle (4, 6)? −2. Use pencil or black ink to draw a line with this slope through the point (4, 6). (Try to make this graph fairly neat and precise, since details will matter.) The line you just drew is the tangent line to the consumer’s indifference curve at the point (4, 6). (b) The indifference curve through the point (4, 6) passes through the points ( 10 ,0), (7, 2 ), and (2, 12 ). Use blue ink to sketch in this indifference curve. Incidentally, the equation for Burt’s indifference curve through the point (4, 6) is x2 = 72/(x1 + 2)−6. 0 4 8 12 16 4 8 12 Cookies Glasses of milk 16 a b Red Line Black Line Blue curve (c) Burt currently has the bundle (4, 6). Ernie offers to give Burt 9 glasses of milk if Burt will give Ernie 3 cookies. If Burt makes this trade, he would have the bundle (1, 15). Burt refuses to trade. Was this a wise decision? Yes, U(1, 15) = 63 < U(4, 6) = 72. Mark the bundle (1, 15) on your graph. (d) Ernie says to Burt, “Burt, your marginal rate of substitution is −2. That means that an extra cookie is worth only twice as much to you as an extra glass of milk. I offered to give you 3 glasses of milk for every cookie you give me. If I offer to give you more than your marginal rate of substitution, then you should want to trade with me.” Burt replies
"Ernie, you are right that my marginal rate of substitution is -2. That means that i am willing to make small trades where i get more than 2 glasses of milk for every cookie I give you, but 9 glasses of milk for 3 cookies is too big a trade. My indifference curves are not straight lines you see. Would Burt be willing to give up l cookie for 3 glasses of milk? Yes, U(3, 9)=75>U(4, 6)=72. Would Burt object to giving up 2 cookies for 6 glasses of milk? No, U(2, 12)=72= U(4,6). (e) On your graph, use red ink to draw a line with slope -3 through the point(4, 6). This line shows all of the bundles that Burt can achieve by trading cookies for milk(or milk for cookies) at the rate of 1 cookie for every 3 glasses of milk. Only a segment of this line represents trades that make Burt better off than he was without trade. Label this line segment on your graph AB 4.4(0)Phil Rupp's utility function is U(a, y)=max, 2y h (a)On the graph below, use blue ink to draw and label the line whose equation is a 10. Also use blue ink to draw and label the line whose (b)If a U(r, y)=10. If x 10 and 2y 10 (c) Now use red ink to sketch in the indifference curve along which U(a, y)=10. Does Phil have convex preferences? No difference
NAME 39 “Ernie, you are right that my marginal rate of substitution is −2. That means that I am willing to make small trades where I get more than 2 glasses of milk for every cookie I give you, but 9 glasses of milk for 3 cookies is too big a trade. My indifference curves are not straight lines, you see.” Would Burt be willing to give up 1 cookie for 3 glasses of milk? Yes, U(3, 9) = 75 > U(4, 6) = 72. Would Burt object to giving up 2 cookies for 6 glasses of milk? No, U(2, 12) = 72 = U(4, 6). (e) On your graph, use red ink to draw a line with slope −3 through the point (4, 6). This line shows all of the bundles that Burt can achieve by trading cookies for milk (or milk for cookies) at the rate of 1 cookie for every 3 glasses of milk. Only a segment of this line represents trades that make Burt better off than he was without trade. Label this line segment on your graph AB. 4.4 (0) Phil Rupp’s utility function is U(x,y) = max{x, 2y}. (a) On the graph below, use blue ink to draw and label the line whose equation is x = 10. Also use blue ink to draw and label the line whose equation is 2y = 10. (b) If x = 10 and 2y < 10, then U(x,y) = 10. If x < 10 and 2y = 10, then U(x,y) = 10. (c) Now use red ink to sketch in the indifference curve along which U(x,y) = 10. Does Phil have convex preferences? No. 0 5 10 15 20 5 10 15 x y 20 Red indifference curve Blue lines x=10 2y=10
40 UTILITY (Ch. 4) 4.5(0) As you may recall, Nancy Lerner is taking Professor Stern's economics course. She will take two examinations in the course and her score for the course is the minimum of the scores that she gets on the two exams. Nancy wants to get the highest possible score for the course (a) Write a utility function that represents Nancy's preferences over al- ternative combinations of test scores i and t2 on tests 1 and re- spectively. U(a1, 2)=minL1, 2, or any monotonic transformation 4.6(0)Remember Shirley Sixpack and Lorraine Quiche from the last chapter? Shirley thinks a 16-ounce can of beer is just as good as two B-ounce cans. Lorraine only drinks 8 ounces at a time and hates stale beer, so she thinks a 16-ounce can is no better or worse than an 8-ounce (a) Write a utility function that represents Shirley's preferences between commodity bundles comprised of 8-ounce cans and 16-ounce cans of beer Let X stand for the number of 8-ounce cans and y stand for the number of 16-ounce cans u(X,Y)=X+2Y (b)Now write a utility function that represents Lorraine's preferences X,y=X+y (c)Would the function utility U(X, Y=100X+200Y represent Shirley's preferences? Yes. Would the utility function U(c, y)=(5X +10Y)2 represent her preferences? Yes. Would the utility function U(, y) X+3Y represent her preferences? No (d) Give an example of two commodity bundles such that Shirley likes the first bundle better than the second bundle, while lorraine likes the second bundle better than the first bundle. Shirley prefer (0, 2) to (3,0). Lorraine disagrees 4.7(0) Harry Mazzola has the utility function u(a1, 22)= min ai+ C 2, 2. 1+a2, where aI is his consumption of corn chips and 2 is his consumption of french fries (a)On the graph below, use a pencil to draw the locus of points along which 1+2 2=2 1 +12. Use blue ink to show the locus of points for which 1+2. C2 12, and also use blue ink to draw the locus of points or which 2 T1+a
40 UTILITY (Ch. 4) 4.5 (0) As you may recall, Nancy Lerner is taking Professor Stern’s economics course. She will take two examinations in the course, and her score for the course is the minimum of the scores that she gets on the two exams. Nancy wants to get the highest possible score for the course. (a) Write a utility function that represents Nancy’s preferences over alternative combinations of test scores x1 and x2 on tests 1 and 2 respectively. U(x1,x2) = min{x1,x2}, or any monotonic transformation. 4.6 (0) Remember Shirley Sixpack and Lorraine Quiche from the last chapter? Shirley thinks a 16-ounce can of beer is just as good as two 8-ounce cans. Lorraine only drinks 8 ounces at a time and hates stale beer, so she thinks a 16-ounce can is no better or worse than an 8-ounce can. (a) Write a utility function that represents Shirley’s preferences between commodity bundles comprised of 8-ounce cans and 16-ounce cans of beer. Let X stand for the number of 8-ounce cans and Y stand for the number of 16-ounce cans. u(X,Y ) = X + 2Y . (b) Now write a utility function that represents Lorraine’s preferences. u(X,Y ) = X + Y . (c) Would the function utility U(X,Y ) = 100X+200Y represent Shirley’s preferences? Yes. Would the utility function U(x,y) = (5X + 10Y )2 represent her preferences? Yes. Would the utility function U(x,y) = X + 3Y represent her preferences? No. (d) Give an example of two commodity bundles such that Shirley likes the first bundle better than the second bundle, while Lorraine likes the second bundle better than the first bundle. Shirley prefers (0,2) to (3,0). Lorraine disagrees. 4.7 (0) Harry Mazzola has the utility function u(x1,x2) = min{x1 + 2x2, 2x1 + x2}, where x1 is his consumption of corn chips and x2 is his consumption of french fries. (a) On the graph below, use a pencil to draw the locus of points along which x1 + 2x2 = 2x1 + x2. Use blue ink to show the locus of points for which x1 + 2x2 = 12, and also use blue ink to draw the locus of points for which 2x1 + x2 = 12