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1 Preferences M. Utku Ünver Micro Theory Boston College M. Utku Ünver Micro Theory (BC) Preferences 1 / 20

2 Preference Relations Given any two consumption bundles x = (x 1, x 2 ) and y = (y 1, y 2 ), the consumers can rank them as to their desirability using his preference relations: (x 1, x 2 ) (y 1, y 2 ): Bundle (x 1, x 2 ) is strictly preferred to (y 1, y 2 ). (x 1, x 2 ) (y 1, y 2 ): Bundle (x 1, x 2 ) is indifferent to (y 1, y 2 ). (x 1, x 2 ) (y 1, y 2 ): Bundle (x 1, x 2 ) is weakly preferred to (y 1, y 2 ) or (x 1, x 2 ) (y 1, y 2 ): Bundle (x 1, x 2 ) is at least as good as (y 1, y 2 ). These are related relations : x y and y x x y x y and y x x y When we talk about the preference relation or preferences, we usually mean the at-least-as-good-as relation. Other relations can be derived from it. M. Utku Ünver Micro Theory (BC) Preferences 2 / 20

3 Assumptions about preferences Completeness: It is always possible to rank-order any two bundles x and y by preference. Therefore we either have x y or y x or both. Reflexivity: Any bundle is at least as good as itself. i. e. x x. Transitivity: If x is at least as good as y and y is at least as good as z, then x is at least as good as z, i.e. x y and y z x z. M. Utku Ünver Micro Theory (BC) Preferences 3 / 20

4 Example 1 Question: Johnny goes to Ben and Jerry s ice cream parlor. There are seven menu items regarding the alternatives at the shop: (The bundles are denoted as (x 1, x 2 ) where x 1 is the scoops of ice cream and x 2 is the pumps of chocolate syrup.) (3,1), (2,2), (5,1),(4, 1 2 ),(8, 1 4 ),(1,2),( 1 2,10). We know that (3, 1) (5, 1), (5, 1) (3, 1); (5, 1) (8, 1 4 ), (8, 1 4 ) (5, 1); (1, 2) (3, 1), (3, 1) (1, 2); (2, 2) (1, 2), (1, 2) (2, 2); (4, 1 2 ) ( 1 2, 10), ( 1 2, 10) (4, 1 2 ); ( 1 2, 10) (8, 1 4 ), (8, 1 4 ) ( 1 2, 10) If his preferences are transitive, how can he rank his choices? Are these preferences complete? M. Utku Ünver Micro Theory (BC) Preferences 4 / 20

5 Answer: (2, 2) (1, 2) (3, 1) (5, 1) (8, 1 4 ) (1 2, 10) (4, 1 2 ) (1 2, 10) As, we can compare any pair of bundles with respect to weak preference relation, his preferences are complete. M. Utku Ünver Micro Theory (BC) Preferences 5 / 20

6 Example 2 Question: John consumes his coffee with sugar. He prefers sweeter coffee. However he cannot taste differences in sugar if it is less then 1 tea spoon. For example, he is indifferent between (1 cup of coffee, 1 2 tsp. sugar) and (1 cup of coffee, 2 3 tsp. sugar) since Are his preferences transitive? Answer: No. Take these three bundles; z=(1 cup of coffee, 3 2 tsp. sugar), y= (1 cup of coffee, 1 tsp. sugar) and x=(1 cup of coffee, 1 4 tsp. sugar). He weakly prefers x to y (since he is indifferent). Also he weakly prefers y and z (since he is indifferent). However he does not weakly prefer x to z (since he strictly prefers z to x). M. Utku Ünver Micro Theory (BC) Preferences 6 / 20

7 Indifference Curves It is convenient to describe preferences graphically using a construction known as indifference curves. An indifference curve is the set (or curve) of bundles that are indifferent to each other. Indifference curve at (x 1, x 2 ) M. Utku Ünver Micro Theory (BC) Preferences 7 / 20

8 Indifference Curves-2 z y x M. Utku Ünver Micro Theory (BC) Preferences 8 / 20

9 Indifference Curves-3 Indifference curves of a single consumer cannot intersect each other. Why? Suppose they did: x y x z but y z; contradicting transitivity of M. Utku Ünver Micro Theory (BC) Preferences 9 / 20

10 Examples of Preferences Perfect Substitutes: The consumers are willing to substitute one good for another at a constant rate. The simplest case is one to one basis. The indifference curves have constant slopes and moreover they all have the same slope. The slope does not need to be -1, that is: these items need not be substituted at one-to-one basis! M. Utku Ünver Micro Theory (BC) Preferences 10 / 20

11 Perfect Complements: Goods that are always consumed together in fixed proportions. For example left shoe, right shoe. Indifference Curves These proportions do not need to be one-to-one! M. Utku Ünver Micro Theory (BC) Preferences 11 / 20

12 Bads: A commodity that a consumer does not like. Example indifference curves for One Good and One Bad M. Utku Ünver Micro Theory (BC) Preferences 12 / 20

13 Neutrals: A commodity that a consumer does not care for one way or another. Indifference curves are always like this for one Neutral and one Good M. Utku Ünver Micro Theory (BC) Preferences 13 / 20

14 Satiation: There is an overall best bundle for the consumer and the closer he is to the best bundle, the better off he is. Example: M. Utku Ünver Micro Theory (BC) Preferences 14 / 20

15 Properties of Preferences In general it will be convenient to focus on a few general shapes of indifference curves. We will often assume that more is better. (i.e. we will be talking about goods rather than bads.) Monotonic Preferences: If x = (x 1, x 2 ) and y = (y 1, y 2 ) are such that x 1 y 1, x 2 y 2 and x = y, then (x 1, x 2 ) (y 1, y 2 ). M. Utku Ünver Micro Theory (BC) Preferences 15 / 20

16 Monotonicity implies that indifference curves are negatively sloped (or zero sloped). M. Utku Ünver Micro Theory (BC) Preferences 16 / 20

17 Convex Preferences: We will often assume that averages are weakly preferred to extremes. Example: (little food, a lot of water) could be indifferent to (a lot of food, little water). But both are worse than (some food, some water). Definition: Suppose (x 1, x 2 ) (y 1, y 2 ) and t (0, 1). Then (tx 1 + (1 t)y 1, tx 2 + (1 t)y 2 ) (x 1, x 2 ) M. Utku Ünver Micro Theory (BC) Preferences 17 / 20

18 The Marginal Rate of Substitution (MRS) The slope of an indifference curve is known as the marginal rate of substitution (MRS). MRS measures the rate at which the consumer is willing to substitute one good for another. Here x 2 x 1 is the rate at which the consumer is willing to substitute good 2 for good 1. The slope is negative. MRS is usually a negative number. M. Utku Ünver Micro Theory (BC) Preferences 18 / 20

19 Trade and MRS Suppose that a consumer consuming (x 1, x 2 ) given an opportunity to trade at a rate E. He will prefer staying put only if E is equal to MRS Illustration with a picture: M. Utku Ünver Micro Theory (BC) Preferences 19 / 20

20 Properties of Preferences and MRS Monotonic Preferences MRS < 0. Strictly convex preferences diminishing marginal rate of substitution. That is: as we increase x 1, the slope of the indifference curve decreases in absolute value. This means the amount of good 2 the person is willing to substitute for good 1 is decreasing as as you increase the amount of good 1 consumed. M. Utku Ünver Micro Theory (BC) Preferences 20 / 20

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