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1.4: Preferences I: Indifference Curves

ECON 306 · Microeconomic Analysis · Spring 2020

Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/microS20
microS20.classes.ryansafner.com

Consumer's Objectives

  • What do consumers want? What do they maximize?

  • Avoid being normative & make as few assumptions as possible

  • We'll assume people maximize preferences

    • WTF does that mean?

Preferences

Preferences I

  • Which bundles of (x,y) are preferred over others?

Example:

a=(412) or b=(612)

Preferences II

  • We will allow three possible answers:

Preferences II

  • We will allow three possible answers:
  1. a≻b: Strictly prefer a over b

Preferences II

  • We will allow three possible answers:

  1. a≻b: Strictly prefer a over b

  2. a≺b: Strictly prefer b over a

Preferences II

  • We will allow three possible answers:

  1. a≻b: Strictly prefer a over b

  2. a≺b: Strictly prefer b over a

  3. a∼b: Indifferent between a and b

Preferences II

  • We will allow three possible answers:

  1. a≻b: Strictly prefer a over b

  2. a≺b: Strictly prefer b over a

  3. a∼b: Indifferent between a and b

  • Preferences are a list of all such comparisons between all bundles

Assumptions About Preferences

  • We assume preferences are "well-behaved" to model:

Assumptions About Preferences

  • We assume preferences are "well-behaved" to model:
  1. Reflexivity : any bundle is at least as preferred as itself

Assumptions About Preferences

  • We assume preferences are "well-behaved" to model:

  1. Reflexivity : any bundle is at least as preferred as itself

  2. Completeness : any two bundles can be compared

Assumptions About Preferences

  • We assume preferences are "well-behaved" to model:

  1. Reflexivity : any bundle is at least as preferred as itself

  2. Completeness : any two bundles can be compared

  3. Transitivity : rankings are logically consistent:

    • If a≻b and b≻c, then a≻c

Assumptions About Preferences

  • Are these good assumptions?

  • Typical in economics: very often yes, sometimes no!

  • See Behavioral economics for interesting anomalies and exceptions

Indifference Curves

Mapping Preferences Graphically I

  • For each bundle, we now have 3 pieces of information:

    • amount of x
    • amount of y
    • preference compared to other bundles

  • How to represent this information graphically?

Mapping Preferences Graphically II

  • Cartographers have the answer for us

  • On a map, contour lines link areas of equal height

  • We will use "indifference curves" to link bundles of equal preference

Mapping Preferences Graphically III

3-D "Mount Utility"

WebGL is not supported by your browser - visit https://get.webgl.org for more info

02468

2-D Indifference Curve Contours

Indifference Curves: Example

Example: Suppose you are hunting for an apartment. You value both the size of the apartment and the number of friends that live nearby.

Indifference Curves: Example

Example: Suppose you are hunting for an apartment. You value both the size of the apartment and the number of friends that live nearby.

  • Apt. A has 1 friend nearby and is 1,200 ft2

Indifference Curves: Example

Example: Suppose you are hunting for an apartment. You value both the size of the apartment and the number of friends that live nearby.

  • Apt. A has 1 friend nearby and is 1,200 ft2
    • Apartments that are larger and/or have more friends ≻A

Indifference Curves: Example

Example: Suppose you are hunting for an apartment. You value both the size of the apartment and the number of friends that live nearby.

  • Apt. A has 1 friend nearby and is 1,200 ft2
    • Apartments that are larger and/or have more friends ≻A
    • Apartments that are smaller and/or have fewer friends ≺A

Indifference Curves: Example

Example:

  • Apt. A has 1 friend nearby and is 1,200 ft2

  • Apt. B has more friends but less ft2

Indifference Curves: Example

Example:

  • Apt. A has 1 friend nearby and is 1,200 ft2

  • Apt. B has more friends but less ft2

  • Apt. C has still more friends but less ft2

Indifference Curves: Example

Example:

  • Apt. A has 1 friend nearby and is 1,200 ft2

  • Apt. B has more friends but less ft2

  • Apt. C has still more friends but less ft2

  • If A∼B∼C, on same indifference curve

Indifference Curves: Example

  • Indifferent between all apartments on the same curve

Indifference Curves: Example

  • Indifferent between all apartments on the same curve

  • Apts above curve are preferred over apts on curve

    • D≻A∼B∼C
    • On a higher curve

Indifference Curves: Example

  • Indifferent between all apartments on the same curve

  • Apts above curve are preferred over apts on curve

    • D≻A∼B∼C
    • On a higher curve
  • Apts below curve are less preferred than apts on curve
    • E≺A∼B∼C
    • On a lower curve

Assumptions About Indifference Curves

  • Like preferences, indifference curves are "well-behaved" when:

  1. We can always draw indifference curves: two bundles can always be ranked

  2. Monotonic: "more is preferred to less"

  3. Convex: "averages are preferred to extremes"

  4. Indifference curves can never cross: preferences are transitive

Assumption 1: We Can Always Draw Them

  • We can always draw indifference curves: two bundles can always be ranked

  • Every possible bundle (point on graph) is on an indifference curve

Assumption 1: We Can Always Draw Them

  • We can always draw indifference curves: two bundles can always be ranked

  • Every possible bundle (point on graph) is on an indifference curve

Assumption 2: Monotonicity

  • Monotonic: "more is preferred to less"

  • For any bundle b with more of at least one good than bundle a⟹a≺b

  • Moves to NE always preferable

  • Moves to SW always unpreferable

Assumption 2: Monotonicity - Application

Application: What do indifference curves look like for bads?

Assumption 3: Convexity

  • Convex: "averages are preferred to extremes"

  • Take a (weighted) average of any two apartments on curve

  • Any "good balance" of the two goods (e.g. C)≻ "unbalanced" (A or B)

  • People prefer variety in consumption

  • Math: convex function ⟹ line connecting 2 points lies above function

Assumption 3: Convexity - Application

Application: What do non-convex (e.g. concave) indifference curves look like? What kinds of preferences does this imply?

Assumption 4: Curves Never Cross

  • Indifference curves can never cross: preferences are transitive

  • Suppose two curves crossed:

    • A∼B
    • B∼C
    • But C ≻ B!
    • Preferences are not transitive!

Marginal Rate of Substitution

Marginal Rate of Substitution I

  • If I take away one friend nearby, how many more ft2 would you need to keep you indifferent?

Marginal Rate of Substitution I

  • If I take away one friend nearby, how many more ft2 would you need to keep you indifferent?

  • Marginal Rate of Substitution (MRS): rate at which you trade off one good for the other and remain indifferent

  • Think of this as your opportunity cost: # of units of y you need to give up to acquire 1 more x

Marginal Rate of Substitution vs. Budget Constraint Slope

  • Budget constraint (slope) measured the market's tradeoff between x and y based on market prices

  • MRS measures your personal evaluation of x vs. y based on your preferences

  • Foreshadowing: what if they are different? Are you truly maximizing your preferences?

Marginal Rate of Substitution II

  • MRS is the slope of the indifference curve MRSx,y=−ΔyΔx=riserun

  • Amount of y given up for 1 more x

  • Note: slope (MRS) changes along the curve!

Utility

So Where are the Numbers?

  • Long ago (1890s), utility considered a real, measurable, cardinal scale†

  • Utility thought to be lurking in people's brains

    • Could be understood from first principles: calories, water, warmth, etc

  • Obvious problems

† "Neuroeconomics" & cognitive scientists are re-attempting a scientific approach to measure utility

Utility Functions?

  • 20th century innovation: preferences as the objects of maximization

  • We can plausibly measure preferences via implications of peoples' actions!

  • Principle of Revealed Preference: if x and y are both feasible, and if x is chosen over y, then the person must (weakly) prefer x⪰y

  • Flawless? Of course not. But extremely useful!

Utility Functions! I

  • So how can we build a function to "maximize preferences"?

  • Construct a utility function u(⋅)† that represents preference relations (≻,≺,∼)

  • Assign utility numbers to bundles, such that, for any bundles a and b: a≻b⟺u(a)>u(b)

† The ⋅ is a placeholder for whatever goods we are considering (e.g. x, y, burritos, lattes, etc)

Utility Functions! II

  • We can model "as if" the consumer is maximizing utility/preferences by maximizing the utility function:

  • "Maximizing preferences": choosing a such that a≻b for all available b

  • "Maximizing utility": choosing a such that u(a)>u(b) for all available b

  • Identical if they contain the same information

Utility Functions, Pural I

  • Imagine three alternative bundles of (x,y): a=(1,2)b=(2,2)c=(4,3)

Utility Functions, Pural I

  • Imagine three alternative bundles of (x,y): a=(1,2)b=(2,2)c=(4,3)
  • Create a utility function u(⋅) that assigns each bundle a utility level of
u(⋅)
u(a)=1
u(b)=2
u(c)=3

Utility Functions, Pural I

  • Imagine three alternative bundles of (x,y): a=(1,2)b=(2,2)c=(4,3)
  • Create a utility function u(⋅) that assigns each bundle a utility level of
u(⋅)
u(a)=1
u(b)=2
u(c)=3
  • Does it mean that bundle c is 3 times the utility of a?

Utility Functions, Pural II

  • Imagine three alternative bundles of (x,y): a=(1,2)b=(2,2)c=(4,3)
  • Now consider u(⋅) and a second utility function v(⋅):
u(⋅) v(⋅)
u(a)=1 v(a)=3
u(b)=2 v(b)=5
u(c)=3 v(c)=7

Utility Functions, Pural III

  • Utility numbers have an ordinal meaning only, not cardinal

    • Only the ordering c≻b≻a matters!

  • Both are valid:†

    • u(c)>u(b)>u(a)
    • v(c)>v(b)>v(a)

† See the Mathematical Appendix in Today's Class Page for why.

Consumer's Objectives

  • What do consumers want? What do they maximize?

  • Avoid being normative & make as few assumptions as possible

  • We'll assume people maximize preferences

    • WTF does that mean?

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