Comments on Problems. 3.1 This problem offers some practice in deriving utility functions from indifference curve specifications.

Transcription

1 CHAPTER 3 PREFERENCES AND UTILITY These problems provide some practice in eamining utilit unctions b looking at indierence curve maps and at a ew unctional orms. The primar ocus is on illustrating the notion o quasi-concavit (a diminishing MRS) in various contets. The concepts o the budget constraint and utilit maimization are not used until the net chapter. Comments on Problems 3.1 This problem oers some practice in deriving utilit unctions rom indierence curve speciications. 3. This problem ocuses on whether some simple utilit unctions ehibit conve indierence curves. 3.3 This problem shows how utilit unctions can be inerred rom MRS segments. It is a ver simple eample o integrabilit. 3.4 This problem shows that diminishing marginal utilit is not required to obtain a diminishing MRS. All o the unctions are monotonic transormations o one another, so this problem illustrates that diminishing MRS is preserved b monotonic transormations, but diminishing marginal utilit is not. 3.5 In this problem students are asked to provide a ormal, utilit-based eplanation or a variet o advertising slogans. The purpose is to get students to think mathematicall about everda epressions. 3.6 Introduces the ormal deinition o quasi-concavit (rom Chapter ) to be applied to the unctions studied graphicall in Problem This problem is an eploration o the ied-proportions utilit unction. The problem also shows how the goods in such problems can be treated as a composite commodit. 3.8 This problem requires students to graph indierence curves or a variet o unctions, some o which are not quasi-concave. Analtical Problems 3.9 Independent marginal utilities. Shows how analsis can be simpliied i the cross partials o the utilit unction are zero. 9 This edition is intended or use outside o the U.S. onl, with content that ma be dierent rom the U.S. Edition. This ma not be

2 10 Chapter 3: Preerences and Utilit 3.10 Utilit unctions and preerences. The problem asks students to think about how a common phenomenon might be relected in a mathematical unctional orm Cobb-Douglas utilit. Provides some eercises with the Cobb-Douglas unction including how to integrate subsistence levels o consumption into the unctional orm. 3.1 The quasi-linear unction. The problem provides a brie introduction to the quasi-linear orm which (in later chapters) will be used to illustrate a number o interesting outcomes Initial endowments. This problem shows how initial endowments can be treated in simple indierence curve analsis CES utilit. Shows how distributional weights can be incorporated into the CES orm introduced in the chapter without changing the basic conclusions about the unction. δ 3.1 a. U = z. Solutions b. U = + + c. U = + z + z 3. a. The case where the same good is limiting is uninteresting because U( 1, 1) = 1 = k = U(, ) = = U[( 1+ ),( 1+ ) ] = ( 1+ ). I the limiting goods dier, then 1 > 1 = k = <. Hence, ( 1+ ) / > k and ( 1+ ) / > k so the indierence curve is conve. b. Again, the case where the same good is maimum is uninteresting. I the goods dier, 1 < 1 = k = >.( 1+ )/ < k,( 1+ )/< k so the indierence curve is concave, not conve. c. Here ( 1+ 1) = k = ( + ) = [( 1+ )/,( 1+ )/ ] so the indierence curve is neither conve or concave it is linear. This edition is intended or use outside o the U.S. onl, with content that ma be dierent rom the U.S. Edition. This ma not be

3 11 Chapter 3: Preerences and Utilit 3.3 a. The MRS is 1/3 at both points. Since both the points lie on the same indierence curve (as the utilit at both points is the same), this means that the slope o the indierence curve is constant (i.e., straight line). So, the goods are perect substitutes. b. We know that or a Cobb Douglas utilit unction, MRS = ; Using this ormula, the values o the MRS (1/4 at the irst point and at the second) and the values o equations in and that can be solved simultaneousl to get = and = 1. So, the utilit unction is o the ormu = and at the points, we can constuct a pair o c. Yes. There was a redundanc. We never used the inormation that the points were on the same indierence curve. So, to ind out the utilit unction, given the unction is Cobb Douglas, just the MRS and the values o and at the points will suice. (Or, alternativel, some other combination o the inormation in part b ecluding one piece o inormation). 3.4 a. U =, U = 0, U =, U = 0, MRS =. b. c. U = U = U = U = MRS =.,,,, U = U = U = U = MRS = This shows that monotonic transormations ma aect diminishing marginal utilit, but not the MRS. 3.5 a. U( p, b) = p+ b b. U > 0. coke This edition is intended or use outside o the U.S. onl, with content that ma be dierent rom the U.S. Edition. This ma not be

4 1 Chapter 3: Preerences and Utilit c. U ( p, ) > U(0, ) > U(1, ) or p > 1 and all. d. Uk (, ) > Ud (, ) or k = d. e. See the etensions to Chapter Because all o the irst order partials are positive, we must onl check the second order partials. a. = = = Not strictl quasi-concave b., < 0, > 0 Strictl quasi-concave 11 1 c. < 0, = 0, = 0 Strictl quasi-concave 11 1 d. Even i we onl consider cases where, both o the own second order partials are ambiguous and thereore the unction is not necessaril strictl quasi-concave. e., < 0 >0 Strictl quasi-concave a. Uhbmr (,,, ) = Minh (, bm,, r). b. A ull condimented hot dog. c. $1.60 d. $.10 an increase o 31 percent. e. Price would increase onl to $1.75 an increase o 7.8 percent.. Raise prices so that a ull condimented hot dog rises in price to $.60. This would be equivalent to a lump-sum reduction in purchasing power. 3.8 Here we calculate the MRS or each o these unctions: a. MRS = = 31-- MRS is constant. ( ) b. MRS = = = -- MRS is diminishing. ( ) c. MRS = = 1 -- MRS is diminishing This edition is intended or use outside o the U.S. onl, with content that ma be dierent rom the U.S. Edition. This ma not be

5 13 Chapter 3: Preerences and Utilit d. e. MRS ( ) ( ) = = = -- MRS is increasing. ( + ) ( + ) MRS = = = ( + ) ( + ) -- MRS is diminishing. Analtical Problems: 3.9 Independent marginal utilities From problem 3.6, 1 = 0 implies diminishing MRS providing 11, < 0. Conversel, the Cobb-Douglas has 1 > 0, 11, < 0, but also has a diminishing MRS (see problem 3.1a) Utilit unctions and preerences a. Finding such a unctional orm is not eas. One possibilit that partl relects this kind o relationship is: U(, ) = + +. In this case, i <, U(, ) = and this person would bu no. Alternativel, i >, U(, ) = and this person would bu no. b. in this eample, the MRS is discontinuous at =. For <, the MRS is 0. This person would be unwilling to give up an or more. For >, the MRS is ininite now this person is unwilling to give up an or ver large amounts o. c. Depending on consumption levels, either or ma have a zero marginal utilit Cobb-Douglas utilit 1 U/ a. MRS = = = (/) 1 U/ This result does not depend on the sum + which, contrar to production theor, has no signiicance in consumer theor because utilit is unique onl up to a monotonic transormation. b. Mathematics ollows directl rom part a. I > the individual values relativel more highl; hence, 1 d d > or =. c. The unction is homothetic in ( 0) and ( 0), but not in and. This edition is intended or use outside o the U.S. onl, with content that ma be dierent rom the U.S. Edition. This ma not be

6 14 Chapter 3: Preerences and Utilit 3.1 The Quasi-linear Function: a. MRS = b. Check < We have = 1 = 1; = = ; 11 = 0; = ; 1 = 0 So, = = 1 which is negative or >0 c. = C e d. Since the marginal utilit o is a constant at 1, while that o is decreasing as increases as it is o the orm we would epect consumers to shit more towards and awa rom when the get to bu more goods to increase utilit due to an income raise. This is because consumers will alwas tr to maimize utilit and hence, the higher the marginal utilit, the better. e. Reer to Eample 3.4. This unction is usuall used to describe the consumption o one commodit with respect to all other commodities. So, the ln could represent the singular commodit while the represents all the other goods consumed Initial endowments a. b. An trading opportunities that dier rom the MRS at, will provide the opportunit to raise utilit (see igure). This edition is intended or use outside o the U.S. onl, with content that ma be dierent rom the U.S. Edition. This ma not be

7 15 Chapter 3: Preerences and Utilit c. A preerence or the initial endowment will require that trading opportunities raise utilit substantiall. This will be more likel i the trading opportunities and signiicantl dierent rom the initial MRS (see igure) CES utilit δ 1 U/ a. MRS = = = ( /) δ 1 U/ so this unction is homothetic. b. I δ = 1, MRS = /, a constant. I δ = 0, MRS = / (/), which agrees with Problem δ c. For δ < 1 1 δ > 0, so MRS diminishes. d. Follows rom part a, i = MRS = /. e. With =.5, MRS (.9) = δ (.9) =.949 MRS ( 1.1 ) = ( 1.1 ) = 1.05 With δ = 1, MRS (.9) = (.9) =.81 MRS( 1.1 ) = ( 1.1 ) = 1.1 Hence, the MRS changes more dramaticall when δ = -1 than when δ =.5; the lower δ is, the more sharpl curved are the indierence curves. When δ =, the indierence curves are L-shaped impling ied proportions. This edition is intended or use outside o the U.S. onl, with content that ma be dierent rom the U.S. Edition. This ma not be