## BEEM101 Microeconomics

### Problem Set 1: Preferences and utility 微观经济学代考

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1. Exercise – Properties of binary relations (Varian Ch. 3, RQ 2 and 3)

Consider a group of people, Alice, Ben, and Clive. Are the following binary relations on the group complete, reflexive, and transitive?

a) At least as tall as

b) Strictly taller than

a)

• Complete: For any two people, either person A is at least as tall as person B, or B is at least as tall as A, or both.
• Reflexive: Any person is at least as tall as themselves.
• Transitive: For any three people, if A is at least as tall as B and B is at least as tall as C, then A is at least as tall as C.

b)

• Not complete: Two persons from the group could have the same height and then they could not be ordered according to the strictly taller than relation.
• Not reflexive: No person is strictly taller than themselves.
• Transitive: For any three people, if A is strictly taller than B and B is strictly taller than C, then A is strictly taller than C.
1. ### Exercise – Properties of binary relations (Varian Ch. 3, RQ 4) 微观经济学代考

A college football coach says that given any two linemen A and B, he always prefers the one who is bigger and faster. Are his preferences complete and transitive?

• Not complete: We cannot compare two linemen where one is bigger but slower than the other.
• Transitive: For any two linemen, if A is bigger and faster than B and B is bigger and faster than C, then A is bigger and faster than C.
1. Exercise – Properties of binary relations (OR Ch. 1, P 3)

Consider the following preference formation scheme. An individual has in mind a function v : X R that attaches a number to each alternative in X. However, she is sensitive only to significant differences in the value of the function. Specifically, she strictly prefers x to y if v(x)v(y) > 1 and is indifferent between x and y if 1 v(x) v(y) 1. Are the individual’s preferences necessarily transitive?

No, suppose for example v(x) = 1, v(y) = 1.6, and v(z) = 2.1. Then we would have x y and  y z but z x.

1. ### Exercise – Utility representation (OR Ch. 1, P 7)

Let X be the set of all positive integers.

a) An individual prefers the number 8 to all other numbers. Comparing a pair of numbers different from 8 she prefers the higher number. Construct a utility function that represents these preferences.

b) An individual prefers the number 8 to all other numbers. Comparing a pair of numbers different from 8 she prefers the number that is closer to 8. Construct a utility function that represents these preferences.

1. a)
1. b) u(x) = −|x 8|

5.Exercise – Utility representation (Varian Ch. 4, RQ 1 and 2) 微观经济学代考

Suppose u : X R represents an individual’s preference relation. Which of the following transformations f : R R can we apply to u such that the function w : X R defined by w(x) = f(u(x)) represents preferences as well?