Unformatted text preview: 50 CHOICE (Ch. 5) x2 = 20. Therefore we know that the consumer chooses the bundle (x1 , x2 ) = (120, 20). For equilibrium at kinks or at corners, we don’t need the slope of the indiﬀerence curves to equal the slope of the budget line. So we don’t have the tangency equation to work with. But we still have the budget equation. The second equation that you can use is an equation that tells you that you are at one of the kinky points or at a corner. You will see exactly how this works when you work a few exercises. Example: A consumer has the utility function U (x1 , x2 ) = min{x1 , 3x2 }.
The price of x1 is 2, the price of x2 is 1, and her income is 140. Her indiﬀerence curves are Lshaped. The corners of the L’s all lie along the line x1 = 3x2 . She will choose a combination at one of the corners, so this gives us one of the two equations we need for ﬁnding the unknowns x1 and x2 . The second equation is her budget equation, which is 2x1 + x2 = 140. Solve these two equations to ﬁnd that x1 = 60 and x2 = 20. So we know that the consumer chooses the bundle (x1 , x2 ) = (60, 20). When you have ﬁnished these exercises, we hope that you will be able to do the following: • Calculate the best bundle a consumer can aﬀord at given prices and income in the case of simple utility functions where the best aﬀordable bundle happens at a point of tangency. • Find the best aﬀordable bundle, given prices and income for a consumer with kinked indiﬀerence curves. • Recognize standard examples where the best bundle a consumer can aﬀord happens at a corner of the budget set. • Draw a diagram illustrating each of the above types of equilibrium. • Apply the methods you have learned to choices made with some kinds of nonlinear budgets that arise in realworld situations. 5.1 (0) We begin again with Charlie of the apples and bananas. Recall that Charlie’s utility function is U (xA , xB ) = xA xB . Suppose that the price of apples is 1, the price of bananas is 2, and Charlie’s income is 40. (a) On the graph below, use blue ink to draw Charlie’s budget line. (Use a ruler and try to make this line accurate.) Plot a few points on the indiﬀerence curve that gives Charlie a utility of 150 and sketch this curve with red ink. Now plot a few points on the indiﬀerence curve that gives Charlie a utility of 300 and sketch this curve with black ink or pencil. NAME 51 Bananas 40 Red curves 30 Black curve 20 Pencil line 10 Blue budget line e a 0 10 20 30 40 Apples (b) Can Charlie aﬀord any bundles that give him a utility of 150? (c) Can Charlie aﬀord any bundles that give him a utility of 300? Yes. No. (d) On your graph, mark a point that Charlie can aﬀord and that gives him a higher utility than 150. Label that point A. (e) Neither of the indiﬀerence curves that you drew is tangent to Charlie’s budget line. Let’s try to ﬁnd one that is. At any point, (xA , xB ), Charlie’s marginal rate of substitution is a function of xA and xB . In fact, if you calculate the ratio of marginal utilities for Charlie’s utility function, you will ﬁnd that Charlie’s marginal rate of substitution is M RS (xA , xB ) = −xB /xA . This is the slope of his indiﬀerence curve at (xA , xB ). The slope of Charlie’s budget line is −1/2 (give a numerical answer). (f ) Write an equation that implies that the budget line is tangent to an −xB /xA = −1/2. There are indiﬀerence curve at (xA , xB ). many solutions to this equation. Each of these solutions corresponds to a point on a diﬀerent indiﬀerence curve. Use pencil to draw a line that passes through all of these points. 52 CHOICE (Ch. 5) (g) The best bundle that Charlie can aﬀord must lie somewhere on the line you just penciled in. It must also lie on his budget line. If the point is outside of his budget line, he can’t aﬀord it. If the point lies inside of his budget line, he can aﬀord to do better by buying more of both goods. On your graph, label this best aﬀordable bundle with an E . This happens where xA = 20 and xB = 10. Verify your answer by solving the two simultaneous equations given by his budget equation and the tangency condition. (h) What is Charlie’s utility if he consumes the bundle (20, 10)? 200. (i) On the graph above, use red ink to draw his indiﬀerence curve through (20,10). Does this indiﬀerence curve cross Charlie’s budget line, just touch it, or never touch it? Just touch it. 5.2 (0) Clara’s utility function is U (X, Y ) = (X + 2)(Y + 1), where X is her consumption of good X and Y is her consumption of good Y . (a) Write an equation for Clara’s indiﬀerence curve that goes through the 36 point (X, Y ) = (2, 8). Y = X +2 − Clara’s indiﬀerence curve for U = 36.
Y 16 U=36 12 11 8 1. On the axes below, sketch 4 0 4 8 11 12 16 X (b) Suppose that the price of each good is 1 and that Clara has an income of 11. Draw in her budget line. Can Clara achieve a utility of 36 with this budget? Yes. NAME 53 (c) At the commodity bundle, (X, Y ), Clara’s marginal rate of substitution is Y − X+1 . +2 (d) If we set the absolute value of the MRS equal to the price ratio, we have the equation Y +1 X +2 = 1. X + Y = 11. (e) The budget equation is (f ) Solving these two equations for the two unknowns, X and Y , we ﬁnd X= 5 and Y = 6. 5.3 (0) Ambrose, the nut and berry consumer, has a utility function √ U (x1 , x2 ) = 4 x1 + x2 , where x1 is his consumption of nuts and x2 is his consumption of berries. (a) The commodity bundle (25, 0) gives Ambrose a utility of 20. Other points that give him the same utility are (16, 4), (9, 8 ), (4, ), (1, 16 ), and (0, 20 ). Plot these points on the axes below and draw a red indiﬀerence curve through them. 12 (b) Suppose that the price of a unit of nuts is 1, the price of a unit of berries is 2, and Ambrose’s income is 24. Draw Ambrose’s budget line with blue ink. How many units of nuts does he choose to buy? 16 units.
(c) How many units of berries? 4 units. (d) Find some points on the indiﬀerence curve that gives him a utility of 25 and sketch this indiﬀerence curve (in red). (e) Now suppose that the prices are as before, but Ambrose’s income is 34. Draw his new budget line (with pencil). How many units of nuts will he choose? 16 units. How many units of berries? 9 units. 54 CHOICE (Ch. 5) Berries 20 15 10 Red curve 5 Blue line 0 5 10 15 20 Blue line Red curve Pencil line 25 30 Nuts (f ) Now let us explore a case where there is a “boundary solution.” Suppose that the price of nuts is still 1 and the price of berries is 2, but Ambrose’s income is only 9. Draw his budget line (in blue). Sketch the indiﬀerence curve that passes through the point (9, 0). What is the slope of his indiﬀerence curve at the point (9, 0)? −2/3. −1/2. (g) What is the slope of his budget line at this point? (h) Which is steeper at this point, the budget line or the indiﬀerence curve? Indifference curve. No. (i) Can Ambrose aﬀord any bundles that he likes better than the point (9, 0)? 5.4 (1) Nancy Lerner is trying to decide how to allocate her time in studying for her economics course. There are two examinations in this course. Her overall score for the course will be the minimum of her scores on the two examinations. She has decided to devote a total of 1,200 minutes to studying for these two exams, and she wants to get as high an overall score as possible. She knows that on the ﬁrst examination if she doesn’t study at all, she will get a score of zero on it. For every 10 minutes that she spends studying for the ﬁrst examination, she will increase her score by one point. If she doesn’t study at all for the second examination she will get a zero on it. For every 20 minutes she spends studying for the second examination, she will increase her score by one point. NAME 55 (a) On the graph below, draw a “budget line” showing the various combinations of scores on the two exams that she can achieve with a total of 1,200 minutes of studying. On the same graph, draw two or three “indifference curves” for Nancy. On your graph, draw a straight line that goes through the kinks in Nancy’s indiﬀerence curves. Label the point where this line hits Nancy’s budget with the letter A. Draw Nancy’s indiﬀerence curve through this point.
Score on test 2 80 "L" shaped indifference curves 60 40 a 20 Budget line 0 20 40 60 80 100 120 Score on test 1 (b) Write an equation for the line passing through the kinks of Nancy’s indiﬀerence curves. x1 = x2 . 10x1 + 20x2 = (c) Write an equation for Nancy’s budget line. 1, 200.
(d) Solve these two equations in two unknowns to determine the intersection of these lines. This happens at the point (x1 , x2 ) = (40, 40). (e) Given that she spends a total of 1,200 minutes studying, Nancy will maximize her overall score by spending ﬁrst examination and tion. 400 minutes studying for the 800 minutes studying for the second examina 5.5 (1) In her communications course, Nancy also takes two examinations. Her overall grade for the course will be the maximum of her scores on the two examinations. Nancy decides to spend a total of 400 minutes studying for these two examinations. If she spends m1 minutes studying NAME 57 (e) On the same graph, use blue ink to draw the indiﬀerence curve for Elmer that contains bundles that he likes exactly as well as the bundle (1, 1) and the indiﬀerence curve that passes through the point (16, 5). (f ) On your graph, use black ink to show the locus of points at which Elmer’s indiﬀerence curves have kinks. What is the equation for this curve? x = y2. (g) On the same graph, use black ink to draw Elmer’s budget line when the price of x is 1, the price of y is 2, and his income is 8. What bundle does Elmer choose in this situation? (4,2). y 16 Blue curve 12 Blue curves 8 Black line 4 Chosen bundle (16,5) Black curve 0 4 8 12 16 20 24 x (h) Suppose that the price of x is 10 and the price of y is 15 and Elmer buys 100 units of x. What is Elmer’s income? 1,150. (Hint: At ﬁrst you might think there is too little information to answer this question. But think about how much y he must be demanding if he chooses 100 units of x.) 5.7 (0) Linus has the utility function U (x, y ) = x + 3y . (a) On the graph below, use blue ink to draw the indiﬀerence curve passing through the point (x, y ) = (3, 3). Use black ink to sketch the indiﬀerence curve connecting bundles that give Linus a utility of 6. 58 CHOICE (Ch. 5) Y 16 12 8 Red line Black (3,3) curve 4 Blue curve 12 16 X 0 4 8 (b) On the same graph, use red ink to draw Linus’s budget line if the price of x is 1 and the price of y is 2 and his income is 8. What bundle does Linus choose in this situation? (0,4). (c) What bundle would Linus choose if the price of x is 1, the price of y is 4, and his income is 8? (8,0). 5.8 (2) Remember our friend Ralph Rigid from Chapter 3? His favorite diner, Food for Thought, has adopted the following policy to reduce the crowds at lunch time: if you show up for lunch t hours before or after 12 noon, you get to deduct t dollars from your bill. (This holds for any fraction of an hour as well.) NAME 63 (a) If the state adopts none of the new plans, ﬁnd the expenditure on computers that maximizes the district’s utility subject to its budget constraint. 20,000. (b) If plan A is adopted, ﬁnd the expenditure on computers that maximizes the district’s utility subject to its budget constraint. 23,333. (c) On your graph, sketch the indiﬀerence curve that passes through the point (30,000, 40,000) if plan B is adopted. At this point, which is steeper, the indiﬀerence curve or the budget line? The budget line. (d) If plan B is adopted, ﬁnd the expenditure on computers that maximizes the district’s utility subject to its budget constraint. (Hint: Look at your graph.) 30,000. (e) If plan C is adopted, ﬁnd the expenditure on computers that maximizes the district’s utility subject to its budget constraint. 40,000. (f ) If plan D is adopted, ﬁnd the expenditure on computers that maximizes the district’s utility subject to its budget constraint. 23,333. 5.12 (0) The telephone company allows one to choose between two diﬀerent pricing plans. For a fee of $12 per month you can make as many local phone calls as you want, at no additional charge per call. Alternatively, you can pay $8 per month and be charged 5 cents for each local phone call that you make. Suppose that you have a total of $20 per month to spend. (a) On the graph below, use black ink to sketch a budget line for someone who chooses the ﬁrst plan. Use red ink to draw a budget line for someone who chooses the second plan. Where do the two budget lines cross? (80, 8). 64 CHOICE (Ch. 5) Other goods 16 Pencil curve 12 Blue curve 8 Red line Black line 4 0 20 40 60 80 100 120 Local phone calls (b) On the graph above, use pencil to draw indiﬀerence curves for someone who prefers the second plan to the ﬁrst. Use blue ink to draw an indiﬀerence curve for someone who prefers the ﬁrst plan to the second. 5.13 (1) This is a puzzle—just for fun. Lewis Carroll (18321898), author of Alice in Wonderland and Through the Looking Glass, was a mathematician, logician, and political scientist. Carroll loved careful reasoning about puzzling things. Here Carroll’s Alice presents a nice bit of economic analysis. At ﬁrst glance, it may seem that Alice is talking nonsense, but, indeed, her reasoning is impeccable. “I should like to buy an egg, please.” she said timidly. “How do you sell them?” “Fivepence farthing for one—twopence for two,” the Sheep replied. “Then two are cheaper than one?” Alice said, taking out her purse. “Only you must eat them both if you buy two,” said the Sheep. “Then I’ll have one please,” said Alice, as she put the money down on the counter. For she thought to herself, “They mightn’t be at all nice, you know.” (a) Let us try to draw a budget set and indiﬀerence curves that are consistent with this story. Suppose that Alice has a total of 8 pence to spend and that she can buy either 0, 1, or 2 eggs from the Sheep, but no fractional eggs. Then her budget set consists of just three points. The point where she buys no eggs is (0, 8). Plot this point and label it A. On your graph, the point where she buys 1 egg is (1, 2 3 ). (A farthing is 1/4 4 of a penny.) Plot this point and label it B . (b) The point where she buys 2 eggs is (2, 6). Plot this point and label it C . If Alice chooses to buy 1 egg, she must like the bundle B better than either the bundle A or the bundle C . Draw indiﬀerence curves for Alice that are consistent with this behavior. 68 DEMAND (Ch. 6) case is that the consumer will choose a “boundary solution” where she consumes only one good. At this point, her indiﬀerence curve will not be tangent to her budget line. When a consumer has kinks in her indiﬀerence curves, she may choose a bundle that is located at a kink. In the problems with kinks, you will be able to solve for the demand functions quite easily by looking at diagrams and doing a little algebra. Typically, instead of ﬁnding a tangency equation, you will ﬁnd an equation that tells you “where the kinks are.” With this equation and the budget equation, you can then solve for demand. You might wonder why we pay so much attention to kinky indiﬀerence curves, straight line indiﬀerence curves, and other “funny cases.” Our reason is this. In the funny cases, computations are usually pretty easy. But often you may have to draw a graph and think about what you are doing. That is what we want you to do. Think and ﬁddle with graphs. Don’t just memorize formulas. Formulas you will forget, but the habit of thinking will stick with you. When you have ﬁnished this workout, we hope that you will be able to do the following: • Find demand functions for consumers with CobbDouglas and other similar utility functions. • Find demand functions for consumers with quasilinear utility functions. • Find demand functions for consumers with kinked indiﬀerence curves and for consumers with straightline indiﬀerence curves. • Recognize complements and substitutes from looking at a demand curve. • Recognize normal goods, inferior goods, luxuries, and necessities from looking at information about demand. • Calculate the equation of an inverse demand curve, given a simple demand equation. 6.1 (0) Charlie is back—still consuming apples and bananas. His utility function is U (xA , xB ) = xA xB . We want to ﬁnd his demand function for apples, xA (pA , pB , m), and his demand function for bananas, xB (pA , pB , m). (a) When the prices are pA and pB and Charlie’s income is m, the equation for Charlie’s budget line is pA xA +pB xB = m. The slope of Charlie’s indifference curve at the bundle (xA , xB ) is −M U1 (xA , xB )/M U2 (xA , xB ) = The slope of Charlie’s budget line is −pA /pB . Charlie’s indiﬀerence curve will be tangent to his budget line at the point (xA , xB ) if the following equation is satisﬁed: −xB /xA . pA /pB = xB /xA . NAME 69 (b) You now have two equations, the budget equation and the tangency equation, that must be satisﬁed by the bundle demanded. Solve these two equations for xA and xB . Charlie’s demand function for apm ples is xA (pA , pB , m) = 2p , and his demand function for bananas is A m xB (pA , pB , m) = 2p . B (c) In general, the demand for both commodities will depend on the price of both commodities and on income. But for Charlie’s utility function, the demand function for apples depends only on income and the price of apples. Similarly, the demand for bananas depends only on income and the price of bananas. Charlie always spends the same fraction of his income on bananas. What fraction is this? 1/2. 6.2 (0) Douglas Cornﬁeld’s preferences are represented by the utility function u(x1 , x2 ) = x2 x3 . The prices of x1 and x2 are p1 and p2 . 12 (a) The slope of Cornﬁeld’s indiﬀerence curve at the point (x1 , x2 ) is −2x2 /3x1 .
(b) If Cornﬁeld’s budget line is tangent to his indiﬀerence curve at (x1 , x2 ), then p1 x1 = 2/3. (Hint: Look at the equation that equates the slope p2 x2 of his indiﬀerence curve with the slope of his budget line.) When he is consuming the best bundle he can aﬀord, what fraction of his income does Douglas spend on x1 ? 2/5. (c) Other members of Doug’s family have similar utility functions, but the exponents may be diﬀerent, or their utilities may be multiplied by a positive constant. If a family member has a utility function U (x, y ) = cxa xb where a, b, and c are positive numbers, what fraction of his or her 12 income will that family member spend on x1 ? a/(a+b). 6.3 (0) Our thoughts return to Ambrose and his nuts and berries. Am√ brose’s utility function is U (x1 , x2 ) = 4 x1 + x2 , where x1 is his consumption of nuts and x2 is his consumption of berries. (a) Let us ﬁnd his demand function for nuts. The slope of Ambrose’s indiﬀerence curve at (x1 , x2 ) is − √2 1 . x Setting this slope equal to the slope of the budget line, you can solve for x1 without even using the 2 2p2 . budget equation. The solution is x1 = p
1 NAME 69 (b) You now have two equations, the budget equation and the tangency equation, that must be satisﬁed by the bundle demanded. Solve these two equations for xA and xB . Charlie’s demand function for apm ples is xA (pA , pB , m) = 2p , and his demand function for bananas is A m xB (pA , pB , m) = 2p . B (c) In general, the demand for both commodities will depend on the price of both commodities and on income. But for Charlie’s utility function, the demand function for apples depends only on income and the price of apples. Similarly, the demand for bananas depends only on income and the price of bananas. Charlie always spends the same fraction of his income on bananas. What fraction is this? 1/2. 6.2 (0) Douglas Cornﬁeld’s preferences are represented by the utility function u(x1 , x2 ) = x2 x3 . The prices of x1 and x2 are p1 and p2 . 12 (a) The slope of Cornﬁeld’s indiﬀerence curve at the point (x1 , x2 ) is −2x2 /3x1 .
(b) If Cornﬁeld’s budget line is tangent to his indiﬀerence curve at (x1 , x2 ), then p1 x1 = 2/3. (Hint: Look at the equation that equates the slope p2 x2 of his indiﬀerence curve with the slope of his budget line.) When he is consuming the best bundle he can aﬀord, what fraction of his income does Douglas spend on x1 ? 2/5. (c) Other members of Doug’s family have similar utility functions, but the exponents may be diﬀerent, or their utilities may be multiplied by a positive constant. If a family member has a utility function U (x, y ) = cxa xb where a, b, and c are positive numbers, what fraction of his or her 12 income will that family member spend on x1 ? a/(a+b). 6.3 (0) Our thoughts return to Ambrose and his nuts and berries. Am√ brose’s utility function is U (x1 , x2 ) = 4 x1 + x2 , where x1 is his consumption of nuts and x2 is his consumption of berries. (a) Let us ﬁnd his demand function for nuts. The slope of Ambrose’s indiﬀerence curve at (x1 , x2 ) is − √2 1 . x Setting this slope equal to the slope of the budget line, you can solve for x1 without even using the 2 2p2 . budget equation. The solution is x1 = p
1 70 DEMAND (Ch. 6) (b) Let us ﬁnd his demand for berries. Now we need the budget equation. In Part (a), you solved for the amount of x1 that he will demand. The budget equation tells us that p1 x1 + p2 x2 = M . Plug the solution that you found for x1 into the budget equation and solve for x2 as a function p M of income and prices. The answer is x2 = p − 4 p2 . 2 1 (c) When we visited Ambrose in Chapter 5, we looked at a “boundary solution,” where Ambrose consumed only nuts and no berries. In that example, p1 = 1, p2 = 2, and M = 9. If you plug these numbers into the formulas we found in Parts (a) and (b), you ﬁnd x1 = 16 , and x2 = −3.5 . Since we get a negative solution for x2 , it must be that the budget line x1 + 2x2 = 9 is not tangent to an indiﬀerence curve when x2 ≥ 0. The best that Ambrose can do with this budget is to spend all of his income on nuts. Looking at the formulas, we see that at the prices p1 = 1 and p2 = 2, Ambrose will demand a positive amount of both goods if and only if M > 16. 6.4 (0) Donald Fribble is a stamp collector. The only things other than stamps that Fribble consumes are Hostess Twinkies. It turns out that Fribble’s preferences are represented by the utility function u(s, t) = s + ln t where s is the number of stamps he collects and t is the number of Twinkies he consumes. The price of stamps is ps and the price of Twinkies is pt . Donald’s income is m. (a) Write an expression that says that the ratio of Fribble’s marginal utility for Twinkies to his marginal utility for stamps is equal to the ratio of the price of Twinkies to the price of stamps. 1/t = pt /ps . (Hint: The derivative of ln t with respect to t is 1/t, and the derivative of s with respect to s is 1.) (b) You can use the equation you found in the last part to show that if he buys both goods, Donald’s demand function for Twinkies depends only on the price ratio and not on his income. Donald’s demand function for Twinkies is t(ps , pt , m) = ps /pt . (c) Notice that for this special utility function, if Fribble buys both goods, then the total amount of money that he spends on Twinkies has the peculiar property that it depends on only one of the three variables m, pt , and ps , namely the variable ps . (Hint: The amount of money that he spends on Twinkies is pt t(ps , pt , m).) 76 DEMAND (Ch. 6) (d) Which of these goods satisfy your textbook’s deﬁnition of necessity goods at most income levels? Food prepared at home, housing.
(e) On the graph below, use the information from Table 6.1 to draw “Engel curves.” (Use total expenditure on current consumption as income for purposes of drawing this curve.) Use red ink to draw the Engel curve for food prepared at home. Use blue ink to draw an Engel curve for food away from home. Use pencil to draw an Engel curve for clothing. How does the shape of an Engel curve for a luxury diﬀer from the shape of an Engel curve for a necessity? The curve for a luxury gets flatter as income rises, the curve for a necessity gets steeper.
Total expenditures (thousands of dollars) 12 Blue line 9 Pencil line Red line 6 3 0 750 1500 2250 3000 Expenditure on specific goods 6.10 (0) Percy consumes cakes and ale. His demand function for cakes is qc = m − 30pc + 20pa , where m is his income, pa is the price of ale, pc is the price of cakes, and qc is his consumption of cakes. Percy’s income is $100, and the price of ale is $1 per unit. (a) Is ale a substitute for cakes or a complement? Explain. A substitute. An increase in the price of ale increases demand for cakes. NAME 77 (b) Write an equation for Percy’s demand function for cakes where income and the price of ale are held ﬁxed at $100 and $1. qc = 120 − 30pc . pc = 4 − qc /30.
Use blue ink to draw (c) Write an equation for Percy’s inverse demand function for cakes where income is $100 and the price of ale remains at $1. At what price would Percy buy 30 cakes? Percy’s inverse demand curve for cakes. $3. (d) Suppose that the price of ale rises to $2.50 per unit and remains there. Write an equation for Percy’s inverse demand for cakes. pc = 5 − qc /30.
for cakes. Use red ink to draw in Percy’s new inverse demand curve Price 4 Red Line 3 2 Blue Line 1 0 30 60 90 120 Number of cakes 6.11 (0) Richard and Mary Stout have fallen on hard times, but remain rational consumers. They are making do on $80 a week, spending $40 on food and $40 on all other goods. Food costs $1 per unit. On the graph below, use black ink to draw a budget line. Label their consumption bundle with the letter A. (a) The Stouts suddenly become eligible for food stamps. This means that they can go to the agency and buy coupons that can be exchanged for $2 worth of food. Each coupon costs the Stouts $1. However, the maximum number of coupons they can buy per week is 10. On the graph, draw their new budget line with red ink. 78 DEMAND (Ch. 6) (b) If the Stouts have homothetic preferences, how much more food will they buy once they enter the food stamp program?
Dollars worth of other things 120 5 units. 100 80 60 New consumption point 40 a Red budget line 20 Black budget line 0 20 45 40 60 80 100 120 Food
Calculus 6.12 (2) As you may remember, Nancy Lerner is taking an economics course in which her overall score is the minimum of the number of correct answers she gets on two examinations. For the ﬁrst exam, each correct answer costs Nancy 10 minutes of study time. For the second exam, each correct answer costs her 20 minutes of study time. In the last chapter, you found the best way for her to allocate 1200 minutes between the two exams. Some people in Nancy’s class learn faster and some learn slower than Nancy. Some people will choose to study more than she does, and some will choose to study less than she does. In this section, we will ﬁnd a general solution for a person’s choice of study times and exam scores as a function of the time costs of improving one’s score. (a) Suppose that if a student does not study for an examination, he or she gets no correct answers. Every answer that the student gets right on the ﬁrst examination costs P1 minutes of studying for the ﬁrst exam. Every answer that he or she gets right on the second examination costs P2 minutes of studying for the second exam. Suppose that this student spends a total of M minutes studying for the two exams and allocates the time between the two exams in the most eﬃcient possible way. Will the student have the same number of correct answers on both exams? 132 INTERTEMPORAL CHOICE (Ch. 10) You will also be asked to determine the eﬀects of inﬂation on consumer behavior. The key to understanding the eﬀects of inﬂation is to see what happens to the budget constraint. Example: Suppose that in the previous example, there happened to be
an inﬂation rate of 6%, and suppose that the price of period1 goods is 1. Then if you save $1 in period 1 and get it back with 10% interest, you will get back $1.10 in period 2. But because of the inﬂation, goods in period 2 cost 1.06 dollars per unit. Therefore the amount of period1 consumption that you have to give up to get a unit of period2 consumption is 1.06/1.10 = .964 units of period2 consumption. If the consumer’s money income in each period is unchanged, then his budget equation is c1 + .964c2 = 210. This budget constraint is the same as the budget constraint would be if there were no inﬂation and the interest rate were r , where .964 = 1/(1 + r ). The value of r that solves this equation is known as the real rate of interest. In this case the real rate of interest is about .038. When the interest rate and inﬂation rate are both small, the real rate of interest is closely approximated by the diﬀerence between the nominal interest rate, (10% in this case) and the inﬂation rate (6% in this case), that is, .038 ∼ .10 − .06. As you will see, this is not such a good approximation if inﬂation rates and interest rates are large. 10.1 (0) Peregrine Pickle consumes (c1 , c2 ) and earns (m1 , m2 ) in periods 1 and 2 respectively. Suppose the interest rate is r . (a) Write down Peregrine’s intertemporal budget constraint in present value terms. c1 + c2 (1+r ) = m1 + (1+r) . m2 (b) If Peregrine does not consume anything in period 1, what is the most he can consume in period 2? m1 (1 + r ) + m2 . This is the (future value, present value) of his endowment. Future value. (c) If Peregrine does not consume anything in period 2, what is the most m he can consume in period 1? m1 + (1+2 ) . This is the (future value, r present value) of his endowment. slope of Peregrine’s budget line? Present value. −(1 + r ). What is the 10.2 (0) Molly has a CobbDouglas utility function U (c1 , c2 ) = ca c1−a , 12 where 0 < a < 1 and where c1 and c2 are her consumptions in periods 1 and 2 respectively. We saw earlier that if utility has the form u(x1 , x2 ) = xa x1−a and the budget constraint is of the “standard” form p1 x1 + p2 x2 = 12 m, then the demand functions for the goods are x1 = am/p1 and x2 = (1 − a)m/p2 . NAME 133 (a) Suppose that Molly’s income is m1 in period 1 and m2 in period 2. Write down her budget constraint in terms of present values. c1 + c2 /(1 + r ) = m1 + m2 /(1 + r ). (b) We want to compare this budget constraint to one of the standard form. In terms of Molly’s budget constraint, what is p1 ? is p2 ? 1. What 1/(1 + r ). What is m? m1 + m2 /(1 + r ). (c) If a = .2, solve for Molly’s demand functions for consumption in each period as a function of m1 , m2 , and r . Her demand function for consumption in period 1 is c1 = .2m1 + .2m2 /(1 + r ). Her demand function for consumption in period 2 is c2 = .8(1+ r )m1 + .8m2 . (d) An increase in the interest rate will consumption. It will decrease her period1 increase her period2 consumption and increase her savings in period 1. 10.3 (0) Nickleby has an income of $2,000 this year, and he expects an income of $1,100 next year. He can borrow and lend money at an interest rate of 10%. Consumption goods cost $1 per unit this year and there is no inﬂation. 134 INTERTEMPORAL CHOICE (Ch. 10) Consumption next year in 1,000s 4 3 Red line 2 Blue line 1 a e Squiggly line 0 1 2 3 4 Consumption this year in 1,000s (a) What is the present value of Nickleby’s endowment? $3,000. What is the future value of his endowment? $3,300. With blue ink, show the combinations of consumption this year and consumption next year that he can aﬀord. Label Nickelby’s endowment with the letter E. (b) Suppose that Nickleby has the utility function U (C1 , C2 ) = C1 C2 . Write an expression for Nickleby’s marginal rate of substitution between consumption this year and consumption next year. (Your answer will be a function of the variables C1 , C2 .) MRS = −C2/C1 . (c) What is the slope of Nickleby’s budget line? −1.1. Write an equation that states that the slope of Nickleby’s indiﬀerence curve is equal to the slope of his budget line when the interest rate is 10%. 1.1 = C1 + C2 /C1 . Also write down Nickleby’s budget equation. C2 /1.1 = 3, 000.
(d) Solve these two equations. Nickleby will consume in period 1 and diagram. 1,500 units 1,650 units in period 2. Label this point A on your NAME 135 (e) Will he borrow or save in the ﬁrst period? Save. How much? 500.
(f ) On your graph use red ink to show what Nickleby’s budget line would be if the interest rate rose to 20%. Knowing that Nickleby chose the point A at a 10% interest rate, even without knowing his utility function, you can determine that his new choice cannot be on certain parts of his new budget line. Draw a squiggly mark over the part of his new budget line where that choice can not be. (Hint: Close your eyes and think of WARP.) (g) Solve for Nickleby’s optimal choice when the interest rate is 20%. Nickleby will consume units in period 2. 1,458.3 units in period 1 and 1,750 (h) Will he borrow or save in the ﬁrst period? Save. How much? 541.7.
10.4 (0) Decide whether each of the following statements is true or false. Then explain why your answer is correct, based on the Slutsky decomposition into income and substitution eﬀects. (a) “If both current and future consumption are normal goods, an increase in the interest rate will necessarily make a saver save more.” False. Substitution effect makes him consume less in period 1 and save more. For a saver, income effect works in opposite direction. Either effect could dominate.
(b) “If both current and future consumption are normal goods, an increase in the interest rate will necessarily make a saver choose more consumption in the second period.” True. The income and substitution effects both lead to more consumption in the second period.
10.5 (1) Laertes has an endowment of $20 each period. He can borrow money at an interest rate of 200%, and he can lend money at a rate of 0%. (Note: If the interest rate is 0%, for every dollar that you save, you get back $1 in the next period. If the interest rate is 200%, then for every dollar you borrow, you have to pay back $3 in the next period.) 144 INTERTEMPORAL CHOICE (Ch. 10) (b) If you gave up a unit of consumption goods at the beginning of 1985 and saved your money at interest, you could use the proceeds of your saving to buy 1.05 units of consumption goods at the beginning of 1986. If you gave up a unit of consumption goods at the beginning of 1978 and saved your money at interest, you would be able to use the proceeds of your saving to buy beginning of 1979. .96 units of consumption goods at the 10.12 (1) Marsha Mellow doesn’t care whether she consumes in period 1 or in period 2. Her utility function is simply U (c1 , c2 ) = c1 + c2 . Her initial endowment is $20 in period 1 and $40 in period 2. In an antique shop, she discovers a cookie jar that is for sale for $12 in period 1 and that she is certain she can sell for $20 in period 2. She derives no consumption beneﬁts from the cookie jar, and it costs her nothing to store it for one period. (a) On the graph below, label her initial endowment, E , and use blue ink to draw the budget line showing combinations of period1 and period2 consumption that she can aﬀord if she doesn’t buy the cookie jar. On the same graph, label the consumption bundle, A, that she would have if she did not borrow or lend any money but bought the cookie jar in period 1, sold it in period 2, and used the proceeds to buy period2 consumption. If she cannot borrow or lend, should Marsha invest in the cookie jar? Yes.
(b) Suppose that Marsha can borrow and lend at an interest rate of 50%. On the graph where you labelled her initial endowment, draw the budget line showing all of the bundles she can aﬀord if she invests in the cookie jar and borrows or lends at the interest rate of 50%. On the same graph use red ink to draw one or two of Marsha’s indiﬀerence curves. Period2 consumption 80 60 a 40 e ;
20 Blue line 0 20 Red curves 40 60 80 Period1 consumption NAME 145 (c) Suppose that instead of consumption in the two periods being perfect substitutes, they are perfect complements, so that Marsha’s utility function is min{c1 , c2 }. If she cannot borrow or lend, should she buy the cookie jar? No. If she can borrow and lend at an interest rate of 50%, should she invest in the cookie jar? Yes. If she can borrow or lend as much at an interest rate of 100%, should she invest in the cookie jar? No. True or false 1. F 2. F 3. F 4. F 5. T 6. F Multiple choice 1. b 2. e 3. d 4. c 5. a 6. d 1 ...
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 Fall '08
 Hansen
 Microeconomics, Alice in Wonderland, blue line, Charlie

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