Week Five: Utility Flashcards
Defining utility
What is utility function and levels?
- Construct utility function → assign utility levels to bundles (rankings)
- Utility function → Summarize individual’s preferences
What are the assumptions for utility functions?
- Continuous Function (only cause small changes)
- Reflexive
- Complete
- Transitive
Relations for Utility functions:
Stictly prefer, at least equally prefered, and equally prefered
Preference relation if yields same ranking of alternatives; not units
u()
Ordinal Utility
Define a Utility Function:
Utility Function: assuming a number of every possible consumption bundle such that more-preferred bundle assigned larger number
Define Ordinal Utility:
Represents consumer’s relation → Assigns utility levels to bundles
- Recently referred to as consumer preferences utility is seen as one way to describe preferences
- The concept of “preference” useful for analyzing choice → utility just way describing preferences
Utility Function: assuming a number of every possible consumption bundle such that more-preferred bundle assigned larger number
- Useful for the order of bundles
- Magnitude only relevant to the nature of ranking
- Not unique utility function that represents a preference relation
🧙🏼 Ordinal Utility: Emphasis on ordering bundles
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What is a monotonic transformation?
Monotonic Transformation: transforming one set of numbers into another set in a way that prefers ordr of the numbers.
A monotonic transformation of a utility function → utility fuction that represents same preferences as original utility function.
If u1 > u2 then f(u1) > f(u2)
Utility:
What are utility bundles?
Indifference curve contains equally prefered bundles
Therefore, Equally preferred $\equiv$ Same Utility
- All bundles on indifferences curves have the same utility levels
Define indifference maps
Indifference Maps: collection of all indifference curves given preference relation $\equiv $ utility function
What does a 3d plot of consumption and utility look like?
How do goods, bads, and neutrals relate to utility?
- Good: Increase u as more consumed
- Bad: Decrease u as more consumed
- Neutral: u not change regardless of consumption
Steps to constructing a utility function?
- Specify utility function: U(x_1, x_2)
- Set utility level to constant level k: U(x_1,x_2)=k
This is all the combintions of goods one and two that yield utility level k - Solve for x_2 in previous equations to obtain generic indifference curve
- Give k an arbitrary value and draw curve
Utility function types
Perfect Substitutes:
Utility Function: u(x_1,x_2)=x_1+x_2
Alternatively, v(x_1,x_2)=(x_1+x_2)^2=x_1^2+2x_1x_2+x_2^2$
Suppose substitution rate is 2:1: u(x_1,x_2)=2x_1+x_2
Therefore, generallly:
u(x1,x2) = ax1 + bx2
Utility function types
Perfect Complements
Utility function:
u(x1,x2) = min{ax1, bx2}
Types of Utility Functions
Cobb-Douglas
General:
u(x1,x2) = x1^a x2^(1-a)
Well-behaved curves: (1) monotonic, and (2) convex
What is marginal utility?
Marginal Utility of commodity i rate of change of totel utility as the quantity of commodity i is comsumed changes
- Marginal means “incremental”
- Rate of change is marginal utility with respect to good one.
Note: (use implicit differentiation, because of of the goods are pegged)
Marginal Rate of Substitution
Slope of the indifference curve at particular point (tangent)
Measures willingless of a consumer to exchange one good for another
Utility function used measure MRS for consumption (dx_1, dx_2) that keeps utility constant:
Slope of the indifference curve (marginal rate of substitution) and is the rate at which consumer will just willingly to substitute a small amount of good two for good oen
