Lecture 1: Utility Maximization

Question 1

Utility Maximization

Answer 1

Consumers choose the best bundle of goods (that maximize utility) among the ones that are affordable.

Question 2

Two Primary Utility Maximization Approaches

Answer 2

• The decision maker’s tastes as summarized in her “preference relation”
• Assumptions of Individual Choice Behavior

Question 3

The decision maker’s tastes as summarized in her “preference relation”

Answer 3

The theory is developed by imposing rationality axiom on the decision-makers’ preference and then analyzing the consequences of these preferences for her choice behavior in some set of X alternatives called the consumption set.

Question 4

Individual’s choice behavior

Answer 4

Proceeds by making assumptions directly concerning this behavior

Question 5

Two classic assumptions of individual choice behavior

Answer 5

• (WARP) weak axiom of revealed preferences
• (SARP) Strong Axiom of Revealed Preferences

Question 6

Assumptions of preferences

Answer 6

• Time and space are fixed
• No negatives goods
  *

Question 7

Why are time and space fixed?

Answer 7

• Commodities in different time and spaces are considered different goods (umbrella in Oklahoma v Seattle at various times of the year).

Question 8

The consumption set is denoted as ___

Question 9

We assumed that x is ___ and ___

Answer 9

Closed and Convex

Question 10

A closed set is ___

Answer 10

a set that includes its boundary points.

Question 11

x is closed if ____

Answer 11

every convergent sequence in the set X converges to a point in x.

Question 12

Physical Constraints of Consumption Goods

Answer 12

• Must be consumed in integer amounts
• Fixed space and time for goods
• Consumption set reflecting survival needs

Question 13

The simplest consumption set

Answer 13

Question 14

Example of non-convex set

Answer 14

Question 15

Example of convex set

Answer 15

Question 16

Strict preference relation

Answer 16

Question 17

Answer 17

known as the weak preference relation

Question 18

The indifference relation

Answer 18

Question 19

Standard properties used to order the set of bundles

Answer 19

Question 21

Understanding the complete preference assumption

Answer 21

The consumer must be able to make a choice between the bundles of goods. Indecision will make it impossible to model choices.

Question 22

Understanding the transitivity assumption

Answer 22

this is necessary for rationality and consistency of the consumer’s preferences.

Question 23

Preference relation is rational if

Answer 23

it is complete and transitive

Question 24

Answer 24

The important part is that: rationality of >~ implies both > and ~ is transitive

Question 25

Continuity

Answer 25

Question 26

Why do we need the continuity assumption

Answer 26

It is necessary to rule out certain discontinuities. So we can assume the smoothness of consumer behavior.

If we are working with continuous functions then we can’t find solutions.

Most importantly: if y is strictly preferred to z and if x is a bundle close enough to y, then x must be strictly preferred to z.

Question 27

Utility function

Answer 27

a function, u: x -> R, representing preference relation

Question 28

>~ (math def.)

Answer 28

if for all x,

Question 29

Level curves in utility functions

Answer 29

These are indifference curves.

They represent different combinations of goods that provide the same utility.

Question 30

> (math def.)

Answer 30

Question 31

Weak Monotonicity

Answer 31

~> satisfies if

Thus, there can be cases where vectors can be equal rather than strictly greater.

if x in X, then x >> y then x >y

ex: (3,2) >> (2,2) -> x ~> y

Question 32

Strict Monotonicity

Answer 32

Strong monotonicity says that at least as much of every good and strictly more of some good is strictly better.

Question 33

Euclidian distance between two points

Answer 33

Try a problem until you can do it in your head

Question 34

Local Nonsatitation

Answer 34

A weaker notion that monotonicity

Question 35

Graph of monotonicity and local nonsatiation

Answer 35

Question 36

Convexity

Answer 36

Question 37

Show convexity graphically

Answer 37

Question 38

Strict convexity

Answer 38

Question 39

Convexity implies that an agent prefers ___

How does this translate graphically between strict convexity and normal convexity

Answer 39

averages to extremes

Question 40

Convexity is a generalization of ___

Answer 40

the neoclassical assumption of diminishing marginal rates of substitution.

Question 41

Existence of a utility function: Supposes preferences are ___

Answer 41

• complete,
• reflexive,
• transitive,
• continuous,
• and strongly monotonic.

Question 42

If there exists a utility function, then there exists ___

Answer 42