Representation of Preferences

Transcription

1 Consumer Preference and The Concept Of Utilit Representation of Preferences Bundle/basket a combination of goods and services that an individual might consume. Eample: Bundle A = (60, 30) contains 60 units of food and 30 units of clothing. Consumer preferences: an indication of how a consumer would rank the desirabilit of baskets. Consumer Preference Assumptions 1) Preferences are complete: Abilit to compare, either A >B or B >A, or A~B ( A is indifferent to B ) Preferences must to state that either the consumer prefers one basket to another or he is indifferent. 2) Preferences are transitive: For three bundles A, B, and C, if A >B and B >C, then A >C. If we allow for non transitive preferences, we can show that the must ccle, thus being prove to eploitation. Consumer Preference Assumptions Eample of non transitive preferences: I would rather have an orange than a banana. And I would rather have a banana than an apple. However, (puzzlingl) I would rather have an apple than an orange. 1st 2nd 3rd Orange > Banana Banana > Apple Apple > Orange Orange $1 Banana $1 Apple $1 But now the intransitive consumer is at the original position (owning an orange), and we can start the mone pump again. Individuals with intransitive preferences would then be wiped out of the market. 1

2 Consumer Preferences Assumptions 3) More is better (also referred to as nonsatiation ) A consumer should prefer two hot dogs to one, and three hot dogs to two. This assumption helps rank some of the bundles in the previous figure. A > E > H A > B > J And A > G > D But it doesn t help us rank other bundles: G > E? We don t know. Ranking Sstems Ordinal Ranking allows us to gather information about the order in which a consumer ranks a bundle. We can know that a consumer prefers basket A to B but not how much more we prefers A to B. We onl need ordinal ranking in order to develop all our analsis of consumer theor. Cardinal Ranking allows us to answer the intensit at which a consumer prefers one bundle to another. For eample, I like basket A twice as much as basket B. It is usuall hard for consumers to articulate their intensit. Utilit Functions It measures the level of satisfaction from consuming different bundles. It represents the consumer s preferences. The unit of measurement is Utils. We can think of Utils as a measure of a consumer s happiness in consuming a good, or a bundle of goods. Utilit Functions Eample Eample: A utilit function for a single good. 1/ 2 u( ) Does it meet the above 3 assumptions? 2

3 Utilit Functions Eample Completeness For an two bundles A and B, we must have that either u(a) u(b), or u(a) u(b), or u(a) = u(b), which indicates that either A>B, A<B, or A~B. Hence, Completeness is satisfied since for an two bundles and z, either u() = > z = u(z), or u() = < z = u(z), or u() = = z = u(z). Eample: A a = $4 u(4) = 4 = 2 B b = $9 u(9) = 9 = 3 Utilit Functions Eample Transitivit For eample, for three bundles C=1, B=4, A=5, then u(c) = (1) 1/2 = 1 u(b) = (4) 1/2 = 2 u(a) = (5) 1/2 =2.23 Thus if u(a) > u(b) and u(b) > u(c), then we have that u(a) > u(c). Therefore the preferences are transitive because: if A >B and B >C, then A >C. Note that this propert holds for an three bundles (numbers, since we are talking about quantities of a single good) a, b, and c. In particular, if a > b and b > c then it must be that a > c. Utilit Functions Eample u(a) 5 u(b) 2 u(c) 1 C 1 B 4 5 A u() u(a) u(b) and u(b) u(c) implies; u(a) u(c) Non satiation (more is better) Indeed it is satisfied since u(b) > u(a) for an two bundles in which B is larger than A Eample: A = 1 unit B = 2 units U (1) = 1 = 1 u (2) = 2 > 1. 3

4 Utilit Functions Utilit Functions Eample Marginal Utilit the rate at which utilit changes as consumption of a good increases. Interpretation: how much better off we would be if we received one more unit of good? In mathematical terms, it is the first derivative of the Utilit Function with respect to good. u u( ) That is, the slope of the utilit function, i.e., its rate of growth as we consume more units of the good. Utilit Functions Eample Mathematical Eample If u( ), u 1 1 then As illustrated in the figure u() = is increasing in, but its marginal utilit, u 1 is decreasing in. 2 Utilit Functions Three Important Points when drawing Total and Marginal Utilit Graphs: 1) Total Utilit and marginal utilit cannot be plotted on the same graph, because the variable in the vertical ais differs on each graph. 2) The marginal utilit function is the slope of the total utilit function. 3) The relationship between total and marginal utilit holds for other measures in economics that will be discussed later on in the semester (such as total and marginal product for a firm). 4

5 Utilit Functions Diminishing Marginal Utilit: additional units add less utilit ( U smaller and smaller) Indeed, marginal utilit is decreasing in. Nonetheless, we will assume that U > 0 alwas. That is, more is alwas better, although the last unit consumed is not as good as the first unit consumed (Think about our first glass of water on a hot da and our fourth glass of water). Utilit Functions Utilit Functions with onl one good are ver restrictive. We seek to generalize our analsis to more than one product. Utilit functions with more than one good, e.g, u (, ) Utilit Functions Utilit functions with more than one good, e.g, u(, ) Utilit Functions Utilit functions with more than one good. Mathematical Eample: U (, ) * Point A, (,)=(2,8) provides the same utilit as Point B (,)=(4,4), since the both reach the same height. Let us now find the marginal utilities Amount of U (, ) 1 is fied 2 Amount of U (, ) 1 is fied 2 5

6 Utilit Functions Eercise Learning b doing 3.1 U(,) = Satisfies more is better? Satisfies diminishing marginal utilit? More is better must be checked in two different was: Increase in increase in U Increase in increase in U Alternativel, we can check if > 0 and > 0 for all, > 0. This propert is satisfied since: = is positive for an amount of good and Similarl, = is positive for an amount of good and Utilit Functions Eercise Diminishing marginal utilit just tells us that the additional utilit we get from consuming further amounts of goods is smaller and smaller. The additional utilit we get from consuming additional goods is and. So we need to show: is decreasing in (it turns out to be true) is decreasing in (it turns out to be true) Mathematicall, in order to see whether is decreasing in, we need to check the second derivative of the utilit function with respect to. That is: < 0 and < 0 for all, > / 2 4 for all and / 2 for all and 4 Quer #1 Consider the utilit function U = min ( ). Does good satisf the diminishing marginal utilit propert? A. Yes B. No C. Depending on the number of units of good that the individual consumes. Quer #1 Answer du is which increases in d 6 Indeed, d d d(6) d 6 0 Hence, u(,) = does not satisf the diminishing marginal utilit propert fro good. As a practice, show that u(,) doesn t satisf it for good either. 6

7 What about a different utilit function, such as u(,)=? Indifference Curves Indifference Curves: curves connecting consumption bundles that ield the same level of utilit. Then, = 65 3, which is still increasing in. Therefore, diminishing marginal utilit doesn t hold in this case either. Indifference Curves For utilit function u(, ) = bundles A, B and C ield the same utilit level (and thus lie on the same indifference curve). Graphicall, all three bundles help a consumer reach the same height in his utilit mountain (i.e., the 3D representation of his utilit function we depicted a few slides ago). In particular, u(8,2) = for bundle A, u(4,4) = for bundle B, u(2,8) = for bundle C. Indifference Curves Properties 4 properties of indifference curves (1 of 4): 1. When the consumer likes both goods ( >0, >0), indifference curves are negativel sloped > 0 > 0 Indeed, b the move is better assumption (i.e., > 0 and > 0), points to the northeast of A must be strictl better, while points to the southwest of A must be strictl worse. Hence, none of the shaded areas can lie on the same indifference curve as Point A As a consequence, the indifference curve of bundles ielding the same utilit as bundle A must lie on the unshaded areas. Thus, indifference curves must be negativel sloped. 7

8 Indifference Curves Properties 4 properties of indifference curves (2 of 4): 1. Indifference curves cannot intersect. 2. If the did (as in the figure) then R>Q (impling that U 2 > U 1 ), but S>T (impling that U 1 > U 2 ), which leads to a contradiction. Indifference Curves Properties 4 properties of indifference curves (3 of 4): 1. Ever consumption bundle lies on one and onl one IC. 2. This propert follows from the previous. (We cannot have points like A, which lies on two different I.C.s and thus would have two different utilit values) Indifference Curves Properties 4 properties of indifference curves (4 of 4): 1. ICs are not thick. 2. Otherwise, bundle B would ield the same utilit level as A, which cannot be since B is to the northeast, containing more units of both and, thus it must be strictl preferred to A (b the more is better assumption). Indifference Curves Marginal Rate of Substitution: The rate at which a consumer will give up one good () to get one additional units of another good (), keeping his or her utilit level constant. MRS slope of IC Where is this epression of MRS, coming from? (Proof on the net slide) 8

9 MRS of u(,) First, note that we need to totall differentiate with respect to and (indeed, when moving rightward along an indifference curve, we are simultaneousl varing and, i.e., lowering in order to get more of ). Totall, differentiating, we obtain Since we move along the same indifference utilit is unaffected, i.e., u 0 0 Rearranging, u slope of IC MRS slope of IC u = u/ + u/ MRS of u(,) Diminishing MRS: Initiall ou are willing to give up man glasses of lemonade () for an additional hamburger (). However, when ou have few glasses of lemonade and man hamburgers ou will not be willing to give up as man (or none) glasses of lemonade in order to get an additional hamburger. That is, IC s are bowed in towards the origin. Decreasing slope of I.C. Decreasing MRS, MRS of u(,) A B C u(,) = u 1 unit 1 unit The same increase in one hamburger (good ) leads the consumer to give up man glasses of lemonade when he is at bundle A (consuming a lot of lemonade), but to give up onl a few glasses of lemonade when he is at point C (when he has few glasses of lemonade). D Indifference Curve Application Demand for attributes in cars U(,) where is gas mileage and is horse power MRS, in this contet represents how a tpical consumer is willing to forgo horsepower () in order to get one more unit of gas mileage (). In 1969, MRS, =3.79, while in 1986, such MRS, dropped to 0.71 showing that people are less willing to give up horsepower () in order to get additional units of miles per gallon () 9

10 On average, consumers were willing to forgo 3.79 horsepower to get an additional one mile per gallon in 1969, but are onl willing to forgo.71 horsepower to get an additional one mile per gallon in Indifference Curves Eercise Learning b doing 3.3 U (, ) U (, ) U (, ) a) Draw the IC corresponding to utilit level of U 1 = 128 = 128 in man of its combinations and connect them For eample, G: = 8, = 16 H: = 16, = 8 I: = 32, = 4 Generall, the trick is = 128 = 128/ = 128/8 = 16 = 128/16 = 8 = 128/32 = 4 Point G Point H Point I Indifference Curves Eercise, b) Does I.C. intersect either ais? No, if IC intersected ais, then at the crossing point, = 0, utilit level would be U 1 = 0 128, Similarl, if IC intersected ais, then at the crossing point, = 0, utilit level would be U 1 = Indifference Curves Eercise, c) Does IC indicate that MRS, is diminishing? Yes, since IC is bowed in toward the origin. Note that: MRS so as we move from left to the right ( ), we get smaller ratios, smaller MRS, (diminishing MRS), i.e., the indifference curve becomes flatter. 10

11 Quer #2 Consider the utilit function U =5 +3, which has the following marginal utilities =5 and = 3. The indifference curves for this utilit function: A. Will have a diminishing marginal rate of substitution of for as increases. B. Will have a constant marginal rate of substitution of for as increases. C. Will have an increasing marginal rate of substitution of for as increases. Quer #2 Answer Answer B Taking utilit function U (,)=5+3, and finding the =5 and = 3, we obtain the MRS between goods and is 5/3. Hence, the MRS is constant in Hence, answer B is correct. Eample of an Increasing MRS, U(,)=A 2 +B 2 In this case, =2A and =2B Hence, MRS, 2A 2A A, which increases in B Unlike in previous eamples, these indifference curves are bowed awa from the origin Special Utilit Functions Perfect Substitutes Perfect Substitutes: Aquafina versus Dasani bottled water; or Kingston versus Scandisk memor sticks. Close substitutes: butter versus margarine; or coffee versus black tea. MRS B,M = MRS M,B = constant Eample: Butter (B) and Margarine (M) U = ab + bm, then B = a and M = b Hence, MRS B,M = b a m b Then, slope of IC = a (constant, i.e., independent in ) b Note that if u = ab + am, so b = a and m = a, Then MRSb,m = a/a =1 11

12 Special Utilit Functions Perfect Substitutes Eample: Pancakes and Waffles U = P + 2W, which implies p = 1 and W = 2. A waffle gives him double utilit of a pancake. The consumer is alwas willing to trade 2 pancakes for 1 waffle. MRS p,w p 1 w 2 slope of IC =.5 (It doesn t change in the amount of pancakes) The slope of the indifference curve is constant I.C. are straight lines Hence, the MRS is also constant Special Utilit Functions Perfect Complements Perfect Complements: left shoe and right shoe (consumer wants them in fied proportions), cars and gasoline, or two scoops of peanut butter to one scoop of jell for the perfect PB&J sandwich. U = 10 min(r,l) At G, we have that R=2 and L=2, then U 2 =10*min(2,2) = 10*2=20 At H, we have that R=3 and L=2, then U 2 =10*min(3,2) = 10*2=20 Consumption choices should be allocated at the kink (Wh spend mone on additional units of R if the won t increase our utilit at all?) Special Utilit Functions Perfect Complements Special Utilit Functions Perfect Complements More generall, utilit functions for goods that are perfect complements look like: u ( 1, 2 ) = A min (a 1, b 2 ), where A, a, b > 0 are all positive constants. Slope of IC (MRS) is: Zero at the flat segment of the IC Infinit at the vertical segment of the IC Undefined at the kink of the IC (infinitel man slopes) L Slope = Slope at the kink is undefined Slope = 0 μ 1 R 12

13 Quer #3 Identif the truthfulness of the following statements: I. Diminishing marginal utilit and increasing total utilit are incompatible with each other. II. When Marginal utilit is negative, total utilit is decreasing. a) Both I and II are true b) Both I and II are false c) I is true; II is false d) I is false; II is true Quer #3 Answer Answer D Statement I is false, because total utilit can increase and et eperience diminishing marginal utilit, i.e., in the increasing, but concave, portion of the total utilit, additional increments of utilit become smaller and smaller (diminishing ), but total utilit is still increasing. Statement II is true, since for marginal utilit to become negative, it must be that additional consumption actuall reduces our total utilit, i.e., total utilit is decreasing. For a graphical representation of total and marginal utilit, see slide 13 of this chapter. Special Utilit Functions Cobb Douglas Cobb Douglas Utilit function U(,) and U = are eamples of Cobb Douglas utilit functions Generall, Cobb Douglas utilit functions look like U(,) = A α β where parameters A, α, β > 0 are all positive constants. if α = β = ½, then U = A 1/2 1/2 = A() 1/2 if α = β = 1 and A =1, then U = Special Utilit Functions Cobb Douglas The Douglas utilit function u(,)=a α β satisfies 3 properties: 1) > 0 and > 0. That is, more is better propert holds U (, ) A 1 0 for an and U (, ) A 1 0 for an and 13

14 Special Utilit Functions Cobb Douglas 2. Since > 0 and > 0, then IC is downward slopping (recall propert 1 of ICs) [what about the case in which the consumer dislikes some goods, < 0? You will see that in the Review Session] 3.Diminishing MRS, : 1 A MRS, * * 1 A 1 * * * 1 ( 1) Then movements from left to right ( ) will shrink this ratio, and reduce MRS, Diminishing MRS, (Flatter I.C.) Special Utilit Functions Quasilinear Utilit Quasi linear: Used in economic applications for situations in which the amount of a commodit (such as toothpaste or garlic) doesn t change ver much with income. ICs are parallel displacements to each other Same slope, same MRS between goods, even if income changes. e.g, toothpaste garlic etc. Special Utilit Functions Quasilinear Utilit General Form of Quasilinear utilit: U(,) = ν() + b (linear in, but not linear in ) where ν() is a function that increases in, such as ν() = () 1/2 or ν() = 2. and where b > 0 is a positive constant. If we find the MRS,,, we obtain Special Utilit Functions Quasilinear Utilit General Form of Quasilinear utilit: U(,) = ν() + b (Eample: U(,)= In this case, MRS, is MRS 2 3, For a given =5, MRS, =10/3, which is constant in (as depicted in the previous figure). v ( ) MRS b, That is, for a given value of, the MRS, is constant in. 14

15 Special Utilit Functions Quasilinear Utilit Pet Rock Fad as an eample of quasilinear utilit. In case ou didn t remember what Pet Rocks are surprisingl, the were famous a few ears ago. Special Utilit Functions Quasilinear Utilit Pet Rock Fad Application: observe the differences in the graphs. Pet Rock Fad: all additional income is spent on pet rocks. The fad is almost over: additional income is spent on both goods. 15