Lecture 2 - Utility

Question 1

Perfect substitution indifference curves

Answer 1

All are linear and parallel

Question 2

Perfect complementarity indifference curves

Answer 2

All are right angled with vertices on a ray from the origin

Question 3

Quasi linear indifference curves

Answer 3

Each curve is a vertically shifted copy of the others

Question 4

Cobb Douglas indifference curves

Answer 4

• All curves are hyperbolic asymptoting to but never touching any axis
• Look like the nice convex monotonic indifference curves that we called well behaved indifference curves

Question 5

Marginal Utility

Answer 5

• Tells us how a consumer’s utility changes as we give them a little more of one good
• The marginal utility with respect to good 1 is the rate of change in utility associated with a small change in the amount of good 1 holding the amount of good 2 fixed
• MU depends on the specific utility function and the magnitudes do not tell us anything

Question 6

Typical Assumptions about utility functions (consistent with assumptions on preferences)

Answer 6

1. As long as x and y are goods utility increases as x and/or y increases
   - MU of x is positive
   - MU is the change in utility associated with increasing consumption of x, holding y constant
2. Diminshing marginal utility
   - As consumption of a good increases but at a slower rate (decreasing rate)
3. Utility functions are continous and smooth

Question 7

MRS relationship to MU

Answer 7

The negative of the slope of the indifference curve equals ratio of marginal utilities

Question 8

MRS for Quasi-linear functions

Answer 8

• MRS = - f(x) doesn’t depend on x2
• So the slope of indifference curves for a quasi-linear utility function is constant along any line for which x1 is constant
• Each curve is a vertically shifted copy of the others
• MRS is a constant along any line for which x1 is constant

Question 9

Monotonic transformation and MRS

Answer 9

• Applying a monotonic transformation to a utility function representing a preference relation simply creates another utility function representing the same preference relation

Question 10

Strictly Prefer

Answer 10

The individual definitely wants bundle x compared to bundle y

Question 11

Indifferent

Answer 11

• The individual doesn’t care if they get x or y
• They are just as satisfied with either bundle

Question 12

Weakly Prefers

Answer 12

Bundle x is at least as good as bundle y (indifferent or strictly preferred to y)

Question 13

3 assumptions (axioms) about consistency of consumer preferences

Answer 13

1. Completeness
2. Reflexivity = any bundle x is always at least as preferred as itself
3. Transitivity

Question 14

Good

Answer 14

• When more of a commodity is always preferred, the commodity is a good
• If every commodity is a good then indifference curves are negatively sloped
• If I take away some of good 1, then to keep you indifferent I must give you some of good 2

Question 15

Bad

Answer 15

• A bad is a good that the consumer doesn’t like
• Less of such a commodity is always preferred e.g. pollution
• If I give you more of a bad good then I also must give you more of a ‘good’ good to keep you indifferent or just as satisfied
• Indifference curve would slope up

Question 16

Neutral Good

Answer 16

No matter how much you get of x2 you are just as well off

Question 17

Perfect substitutes

Answer 17

• If a consumer always regards units of commodities 1 and 2 as equivalent then the commodities are perfect substitutes and only the total amount of the 2 commodities in bundles determines their preference rank order
• The consumer is willing to substitute one good for another at a constant rate
• Proportion doesn’t have to be 1 to 1

Question 18

Perfect Complements

Answer 18

• If a consumer always consumes commodities 1 and 2 in fixed proportions then the commodities are perfect complements and only the number of pairs of units of the 2 commodities determines the preferences rank order of bundles
• Proportion doesn’t have to be 1 to 1

Question 19

Monotonicity

Answer 19

• More of any commodity is always preferred
• Non-satiation (satiation point is when we prefer only one point and nothing else)
• Every commodity is a good
• Since it is not unreasonable for an individual to get satiated we are essentially studying situations before the point of satiation is reached
• Implies indifference curves have a negative slope

Question 20

Convexity

Answer 20

Averages are preferred to extremes

Question 21

MRS

Answer 21

• The slope of an indifference curve is its MRS
• MRS is the rate at which the consumer is just willing to substitute one good for another

Question 22

Infinitely divisible

Answer 22

A commodity is infinitely divisible if it can be acquired in any quantity

Question 23

Discrete

Answer 23

A commodity is discrete if it comes in unit lumps of 1,2,3 and so on

Question 24

Ordinal

Answer 24

• Utility is an ordinal concept
• E.g. if U(x) = 6 and U(y) = 2 then the bundle x is strictly preferred to bundle y
• But x is not preferred 3 times as much as is y
• Since only the ranking of the bundles matter there is no unique way to assign utility numbers to the various bundles

Question 25

Monotonic transformation

Answer 25

- A monontonic transformation is a way of transforming one set of numbers into another set of numbers in a way that preserves the order of the numbers - A monotonic transformation of a utility function is a utility function that represents the same preferences as the original utility function - Thus infinite ways to represent a set of preferences

Question 26

Indifference map

Answer 26

- The collection of all indifference curves for a given preference relation is an indifference map - An indifference map is equivalent to a utility function each is the other