(h)Are Coach Steroid's new preferences reflexive? Yes 3.14(0)The Bear family is trying to decide what to have for din ner. Baby Bear says that his ranking of the possibilities is(honey, grubs Goldilocks). Mama Bear ranks the choices(grubs, Goldilocks, honey Thile Papa Bear's ranking is(Goldilocks, honey, grubs). They decide te take each pair of alternatives and let a majority vote determine the famil (a) Papa suggests that they first consider honey vs. grubs, and then the winner of that contest vs. Goldilocks. Which alternative will be chosen? Goldilocks (b) Mama suggests instead that they consider honey vs. Goldilocks and then the winner vs grubs. Which gets chosen? Grubs (c) What order should Baby Bear suggest if he wants to get his favorite food for dinner? Grubs versus goldilocks, then Honey versus the winner (d) Are the Bear family's "collective preferences, " as determined by vot ing, transitive? No 3.15(0)Olson likes strong coffee, the stronger the better. But he cant distinguish small differences. Over the years, Mrs. Olson has discovered hat if she changes the amount of coffee by more than one teaspoon in her six-cup pot, Olson can tell that she did it. But he cannot distinguish ifferences smaller than one teaspoon per pot. Where A and B are tw different cups of coffee, let us write A >B if Olson prefers cup A to cup B. Let us write A> B if Olson either prefers A to B, or can't tell the difference between them. Let us write a n b if olson can't tell the difference between cups A and B. Suppose that Olson is offered cups A B, and C all brewed in the Olsons'six-cup pot. Cup A was brewed using 14 teaspoons of coffee in the pot. Cup b was brewed using 14.75 teaspoons of coffee in the pot and cup C was brewed using 15.5 teaspoons of coffee the pot. For each of the following expressions determine whether it true of false (a)ANB. True (b)BNA. True
NAME 31 (h) Are Coach Steroid’s new preferences reflexive? Yes. 3.14 (0) The Bear family is trying to decide what to have for dinner. Baby Bear says that his ranking of the possibilities is (honey, grubs, Goldilocks). Mama Bear ranks the choices (grubs, Goldilocks, honey), while Papa Bear’s ranking is (Goldilocks, honey, grubs). They decide to take each pair of alternatives and let a majority vote determine the family rankings. (a) Papa suggests that they first consider honey vs. grubs, and then the winner of that contest vs. Goldilocks. Which alternative will be chosen? Goldilocks. (b) Mama suggests instead that they consider honey vs. Goldilocks and then the winner vs. grubs. Which gets chosen? Grubs. (c) What order should Baby Bear suggest if he wants to get his favorite food for dinner? Grubs versus Goldilocks, then Honey versus the winner. (d) Are the Bear family’s “collective preferences,” as determined by voting, transitive? No. 3.15 (0) Olson likes strong coffee, the stronger the better. But he can’t distinguish small differences. Over the years, Mrs. Olson has discovered that if she changes the amount of coffee by more than one teaspoon in her six-cup pot, Olson can tell that she did it. But he cannot distinguish differences smaller than one teaspoon per pot. Where A and B are two different cups of coffee, let us write A B if Olson prefers cup A to cup B. Let us write A B if Olson either prefers A to B, or can’t tell the difference between them. Let us write A ∼ B if Olson can’t tell the difference between cups A and B. Suppose that Olson is offered cups A, B, and C all brewed in the Olsons’ six-cup pot. Cup A was brewed using 14 teaspoons of coffee in the pot. Cup B was brewed using 14.75 teaspoons of coffee in the pot and cup C was brewed using 15.5 teaspoons of coffee in the pot. For each of the following expressions determine whether it is true of false. (a) A ∼ B. True. (b) B ∼ A. True
32 PREFERENCEs (Ch. 3) (c)BNc. True (d)ANC. False (e)CnA. False f)A≥B.True. (g)B≥A.True (h)B≥C.True )A≥C. False )C≥A.True. (((k)A>B. False (B,A. False (m)B>C. False (n)A>C. False (o)C>A. True (p) Is Olson's"at-least-as-good-as"relation, 2, transitive? No (a)Is Olson'scan't-tell-the-difference"relation, N, transitive? No (r)is Olson's"better-than"relation, / transitive. Yes
32 PREFERENCES (Ch. 3) (c) B ∼ C. True. (d) A ∼ C. False. (e) C ∼ A. False. (f) A B. True. (g) B A. True. (h) B C. True. (i) A C. False. (j) C A. True. (k) A B. False. (l) B A. False. (m) B C. False. (n) A C. False. (o) C A. True. (p) Is Olson’s “at-least-as-good-as” relation, , transitive? No. (q) Is Olson’s “can’t-tell-the-difference” relation, ∼, transitive? No. (r) is Olson’s “better-than” relation, , transitive. Yes
Chapter 4 NAME Utility Introduction. In the previous chapter, you learned about preferences and indifference curves. Here we study another way of describing prefer- ences, the utility function. A utility function that represents a persons preferences is a function that assigns a utility number to each commodity bundle. The numbers are assigned in such a way that commodity bundle (a, y) gets a higher utility number than bundle(ar, y) if and only if the consumer prefers (a, y)to(a, y). If a consumer has the utility function U(1, 12), then she will be indifferent between two bundles if they are assigned the same utility If you know a consumer's utility function, then you can find the difference curve passing through any commodity bundle. Recall from the previous chapter that when good 1 is graphed on the horizontal axis and good 2 on the vertical axis, the slope of the indifference curve passing through a point( 1, a2)is known as the marginal rate of substitution. An important and convenient fact is that the slope of an indifference curve is minus the ratio of the marginal utility of good 1 to the marginal utility of ood 2. For those of you who know even a tiny bit of calculus, calculating marginal utilities is easy. To find the marginal utility of either good you just take the derivative of utility with respect to the amount of that good, treating the amount of the other good as a constant. (If you don't know any calculus you can calculate an approximation to marginal utility by the method described in your textbook. Also, at the beginnin of this section of the workbook, we list the marginal utility functions for commonly encountered utility functions. Even if you cant compute these yourself, you can refer to this list when later problems require you to use marginal utilities Example: Arthur's utility function is U(C1, 12)=T1T2. Let us find the indifference curve for Arthur that passes through the point (3, 4). First calculate U(3, 4)=3 4= 12. The indifference curve through this point consists of all (a1, 22) such that T2= 12. This last equation is equivalent to 2= 12/ a1. Therefore to draw Arthur's indifference curve through ( 3, 4), just draw the curve with equation 12= 12/1.At the point(al, 12), the marginal utility of good I is I2 and the marginal utility of good 2 is 1. Therefore Arthur's marginal rate of substitution at the point(3, 4)is-12/a1=-4/ 3 Example: Arthur's uncle, Basil, has the utility function U*(r1, 2) 3 112-10. Notice that U*(1, 2)=3U(1, 12)-10, where U(l, r2)is Arthur's utility function. Since U" is a positive multiple of U minus a con- stant, it must be that any change in consumption that increases U will als increase U*(and vice versa). Therefore we say that Basil's utility function is a monotonic increasing transformation of Arthur's utility function. Let
Chapter 4 NAME Utility Introduction. In the previous chapter, you learned about preferences and indifference curves. Here we study another way of describing preferences, the utility function. A utility function that represents a person’s preferences is a function that assigns a utility number to each commodity bundle. The numbers are assigned in such a way that commodity bundle (x,y) gets a higher utility number than bundle (x0 ,y0 ) if and only if the consumer prefers (x,y) to (x0 ,y0 ). If a consumer has the utility function U(x1,x2), then she will be indifferent between two bundles if they are assigned the same utility. If you know a consumer’s utility function, then you can find the indifference curve passing through any commodity bundle. Recall from the previous chapter that when good 1 is graphed on the horizontal axis and good 2 on the vertical axis, the slope of the indifference curve passing through a point (x1,x2) is known as the marginal rate of substitution. An important and convenient fact is that the slope of an indifference curve is minus the ratio of the marginal utility of good 1 to the marginal utility of good 2. For those of you who know even a tiny bit of calculus, calculating marginal utilities is easy. To find the marginal utility of either good, you just take the derivative of utility with respect to the amount of that good, treating the amount of the other good as a constant. (If you don’t know any calculus at all, you can calculate an approximation to marginal utility by the method described in your textbook. Also, at the beginning of this section of the workbook, we list the marginal utility functions for commonly encountered utility functions. Even if you can’t compute these yourself, you can refer to this list when later problems require you to use marginal utilities.) Example: Arthur’s utility function is U(x1,x2) = x1x2. Let us find the indifference curve for Arthur that passes through the point (3, 4). First, calculate U(3, 4) = 3 × 4 = 12. The indifference curve through this point consists of all (x1,x2) such that x1x2 = 12. This last equation is equivalent to x2 = 12/x1. Therefore to draw Arthur’s indifference curve through (3, 4), just draw the curve with equation x2 = 12/x1. At the point (x1,x2), the marginal utility of good 1 is x2 and the marginal utility of good 2 is x1. Therefore Arthur’s marginal rate of substitution at the point (3, 4) is −x2/x1 = −4/3. Example: Arthur’s uncle, Basil, has the utility function U∗(x1,x2) = 3x1x2 − 10. Notice that U∗(x1,x2)=3U(x1,x2) − 10, where U(x1,x2) is Arthur’s utility function. Since U∗ is a positive multiple of U minus a constant, it must be that any change in consumption that increases U will also increase U∗ (and vice versa). Therefore we say that Basil’s utility function is a monotonic increasing transformation of Arthur’s utility function. Let
34 UTILITY(Ch. 4) us find Basils indifference curve through the point ( 3, 4). First we find lat U(3, 4)=3x3x4-10= 26. The indifference curve passing through this point consists of all (a1, 2) such that 3 01 2-10= 26. Simplify this last expression by adding 10 to both sides of the equation and dividing both sides by 3. You find a1 2= 12, or equivalently, 32= 12/1. This is exactly the same curve as Arthur's indifference curve through (3, 4) We could have known in advance that this would happen, because if tw consumers'utility functions are monotonic increasing transformations of each other, then these consumers must have the same preference relation between any comn nodity bundles When you have finished this workout, we hope that you will be able to do the following: Draw an indifference curve through a specified commodity bundle when you know the utility function Calculate marginal utilities and marginal rates of substitution when you know the utility function. Determine whether one utility function is just a"monotonic transfor mation"of another and know what that implies about preferences Find utility functions that represent preferences when goods are per fect substitutes and when goods are perfect complements Recognize utility functions for commonly studied preferences such as perfect substitutes, perfect complements, and other kinked indiffer ence curves, quasilinear utility, and Cobb-Douglas utility 4.0 Warm Up Exercise. This is the first of severalwarm up ex- ercises"that you will find in Workouts. These are here to help you see how to do calculations that are needed in later problems. The answers t all warm up exercises are in your answer pages. If you find the warm up exercises easy and boring, go ahead--skip them and get on to the main problems. You can come back and look at them if you get stuck later This exercise asks you to calculate marginal utilities and marginal rates of substitution for some common utility functions. These utility functions will reappear in several chapters, so it is a good idea to get to know them now. If you know calculus, you will find this to be a breeze Even if your calculus is shaky or nonexistent, you can handle the first three utility functions just by using the definitions in the textbook. These three are easy because the utility functions are linear. If you do not know any calculus. fill in the rest of the answers from the back of the workbook and keep a copy of this exercise for reference when you encounter these utility functions in later problems
34 UTILITY (Ch. 4) us find Basil’s indifference curve through the point (3, 4). First we find that U∗(3, 4) = 3×3×4−10 = 26. The indifference curve passing through this point consists of all (x1,x2) such that 3x1x2 − 10 = 26. Simplify this last expression by adding 10 to both sides of the equation and dividing both sides by 3. You find x1x2 = 12, or equivalently, x2 = 12/x1. This is exactly the same curve as Arthur’s indifference curve through (3, 4). We could have known in advance that this would happen, because if two consumers’ utility functions are monotonic increasing transformations of each other, then these consumers must have the same preference relation between any pair of commodity bundles. When you have finished this workout, we hope that you will be able to do the following: • Draw an indifference curve through a specified commodity bundle when you know the utility function. • Calculate marginal utilities and marginal rates of substitution when you know the utility function. • Determine whether one utility function is just a “monotonic transformation” of another and know what that implies about preferences. • Find utility functions that represent preferences when goods are perfect substitutes and when goods are perfect complements. • Recognize utility functions for commonly studied preferences such as perfect substitutes, perfect complements, and other kinked indifference curves, quasilinear utility, and Cobb-Douglas utility. 4.0 Warm Up Exercise. This is the first of several “warm up exercises” that you will find in Workouts. These are here to help you see how to do calculations that are needed in later problems. The answers to all warm up exercises are in your answer pages. If you find the warm up exercises easy and boring, go ahead—skip them and get on to the main problems. You can come back and look at them if you get stuck later. This exercise asks you to calculate marginal utilities and marginal rates of substitution for some common utility functions. These utility functions will reappear in several chapters, so it is a good idea to get to know them now. If you know calculus, you will find this to be a breeze. Even if your calculus is shaky or nonexistent, you can handle the first three utility functions just by using the definitions in the textbook. These three are easy because the utility functions are linear. If you do not know any calculus, fill in the rest of the answers from the back of the workbook and keep a copy of this exercise for reference when you encounter these utility functions in later problems
u(a1,I2) MU(a1,I2)MU2(a1, I2) MRS(1, I2) 2r1+3r2 2 3 T1+6x2 4 2/3 +b 2m1+ 1/x1 1/ U’(x1) T12 b b-1 (x1+2)(x2+1 2+1 T1+2 (x1+a)(x2+b)x2+b 1+a i + a2
NAME 35 u(x1,x2) MU1(x1,x2) MU2(x1,x2) MRS(x1,x2) 2x1 + 3x2 2 3 −2/3 4x1 + 6x2 4 6 −2/3 ax1 + bx2 a b −a/b 2 √x1 + x2 1 √x1 1 − 1 √x1 ln x1 + x2 1/x1 1 −1/x1 v(x1) + x2 v0 (x1) 1 −v0 (x1) x1x2 x2 x1 −x2/x1 xa 1xb 2 axa−1 1 xb 2 bxa 1xb−1 2 −ax2 bx1 (x1 + 2)(x2 + 1) x2 + 1 x1 + 2 − x2+1 x1+2 (x1 + a)(x2 + b) x2 + b x1 + a − x2+b x1+a xa 1 + xa 2 axa−1 1 axa−1 2 − x1 x2 a−1