Part 2A. 3. Indifference Curves

Transcription

1 Part 2A. Preferences & Utility Function 3. Indifference Curves 無異曲線 IC and MRS Properties of ICs Conveity, Quasi-Concavity & DMRS Well-Behaved Utility Functions

2 IC and MRS Definition: Indifference Curves The set of all consumption bundles that a consumer views as being equally desirable (indifferent). 2 y Consumption bundle and y provide the same level of utility. U 0 1 2

3 Figure: Bundles of Pizzas and Burritos Lisa Might Consume Preference Maps Graphical interpretation of consumer preferences over goods. Perloff (2014, 3e, GE), Figure 3.1, p.85. 3

4 Figure: The Relationship b/w Utility Function & ICs Perloff (2014, 3e, GE), Figure 3.3, p.90. 4

5 Definition: Indifference Curve Map a complete set of indifference curves that summarize a consumer stastesorpreferences s or preferences. 2 z y U 2 U 1 U 0 1 Perloff (2014, 3e, GE), Figure 3.1(c), p.85. 5

6 Eample: IC, Two-good Case Let U( 1, 2 ) = U 0, U 0 is given. Then totally differentiating the equation gives U U du (, U ) the IC Function This equation gives the IC that 2 as a function of 1. 6

7 Definition: Marginal Rate of Substitution The negative of the slope of the indifference curve at any point is called the marginal rate of substitution (MRS). 2 MRS 12 2 The maimum 0 amount of one good ( 2 ) that a consumer will sacrifice (trade) to obtain one more unit of another U 0 good d( ( 1 ). 1 U U Perloff (2014, 3e, GE), Figure 3.4, p

8 Eample: IC, Two-good Case Let U( 1, 2 ) = U 0, U 0 is given. Then totally differentiating the equation gives U U du MU MU UU 2 0 MU MU MRS MU U U 0 MU 2 MRS is the relative evaluation of two goods. 8

9 MRS changes as and y change. MRS reflects the individual s willingness to trade 2 for 1. 2 MRS12 1 UU 0 2 At point, the IC is steeper. The person would be willing to give up more 2 to gain additional units of 1. y U 1 At point y, the IC is flatter. The person would be willing to give up less 2 to gain additional units of 1. Perloff (2014, 3e, GE), Figure 3.4, p

10 Eample: Suppose an individual s preferences for hamburgers ( 2 ) and soft drinks ( 1 ) can be represented by U(, ) Solving for 2, we get 2 = 100/ 1 2 Solving for MRS, MRS Note: As 1 rises, MRS falls. when 1 = 5, MRS = 4 when 1 =20 20, MRS = U 10 1 Nicholson and Snyder (2012, 11e), Eample 3.1 pp

11 Figure: Indifference Curve for Utility U(, ) At point A (5, 20), the MRS is 4, implying that this person is willing to trade 4y for an additional. At point B (20, 5), however, the MRS is 0.25, implying a greatly reduced willingness to trade. Nicholson and Snyder (2012, 11e), Figure 3.7 p

12 Theorem: Given a utility function, any positive monotonic transformation of it represents the same preference. Proof: Let = (U( )) be a positive monotonic function of a utility function U(), i.e. (U) > 0. MRS U M1 U 1 MU1 M U 2 MU2 U MRS U

13 Eample: U( 1, 2 ) = 1 2 ( U ) U MRS M MU U 12 MRS M MU2 13

14 Properties of ICs IC1: Negatively Sloped (A4: Monotonicity) In the utility function, these i s are assumed to be goods. 2 Preferred to?? More is preferred to less. Economic Goods Worse Than 1 14

15 IC2: Increasing utility to north-east direction ( A4: Monotonicity) Each point must have an IC through it. 2 Utility U 0 < U 1 < U 2 U U 2 U 0 U 1 1 Perloff (2014, 3e, GE), Figure 3.1(c), p

16 IC3: Non-intersection ( A2: Transitivity; A4: Monotonicity ) Q: Can two of an individual s id indifference curves intersect? y ~, z ~ y ~ z. The individual is indifferent between y and. The individual is indifferent between z and. 2 Transitivity suggests that the individual should be indifferent between y and z. z y But y is preferred to z because y contains more 1 and 2 than z. It is a contradiction,. 1 Perloff (2014, 3e, GE), Figure 3.2(c), p

17 IC4: 4 Conve to the origin ( A5: Conveity) 2 Definition: Conveity A set of points is conve if any two points can be joined by a straight line that is contained completely within the set. Well-balanced bundles are preferred to those that are heavily weighted toward one good. 1 17

18 IC5: 5 No Fat ICs ( A4: Monotonicity) ICs cannot be thick. 2 Since and y are on the same IC, ~ y. But y is preferred to because y contains more 1 y and 2 than. It is a contradiction,. 1 Perloff (2014, 3e, GE), Figure 3.2(c), p

19 IC6: Continuous ( A3: Continuity) The continuity of ICs implies that the utility function U(, y) is also continuous

20 Conveity, Quasi-Concavity & DMRS MU 2 1 MRS12 du MU2 2 dmrs 12 d Diminishing MRS 0 U

21 Curvature of Indifference Curves MRS approaches zero (becomes flatter) as we move down and to the right along an IC. This willingness to sacrifice (trade) fewer 2 for one more unit of 1 as we move down and to the right along the IC reflects a diminishing MRS. 21

22 Figure: Curvature of Indifference Curves MRS (willingness to trade) diminishes along many typical indifference curves that are concave to the origin. Different utility functions generate different indifference curves. Perloff (2014, 3e, GE), Figure 3.5, p

23 Definition: Strictly Quasi-Concavity Definition: A function U( 1, 2 ) is strictly quasi-concave if the set {( 1, 2 ) U( 1, 2 ) > U 0 }isaconve set. The utility function U( 1, 2 ) is strictly quasi- concave. 2 d 2 2 dmrs 1 u 11u2 2uu 1 2u12 u22u U 0 Note: DMRS and DMU are two different concepts. dmrs dmu

24 Eample: Let U( 1, 2 ) = Calculating the MUs and MRS: U MU MU MRS12 U U MU MU Differentiating the MU 1 w.r.t. 1 : dmu

25 U Solving for 2, we have the ICs: 2 21 Differentiating the IC wrt w.r.t. U 1 : U MRS g 1 Differentiating the MRS w.r.t. 1 : dmrs U Strictly quasi-concavity: u u 2uu u u u u 1 = 2 2, u 2 = 2 1 u =0 = , u u 12 = u 21 =

26 Well-Behaved Utility Functions Preference: The consumer s preference relation on X is A1: Complete A2: Transitive A3: Continuous A4: Strictly monotonic A5: Strictly conve Utility: The consumer s utility function U: X R is U1: Continuous U2: Strictly increasing U3: Strictly tl quasi- concave U4: At least twice continuously differentiable 26