1.4: Preferences I: Indifference Curves

# 1.4: Preferences I: Indifference Curves
## ECON 306 · Microeconomic Analysis · Fall 2019
### Ryan Safner Assistant Professor of Economics <a href="mailto:safner@hood.edu"> safner@hood.edu</a> <a href="https://github.com/ryansafner/microf19"> ryansafner/microf19</a> <a href="https://microF19.classes.ryansafner.com"> microF19.classes.ryansafner.com</a>

---

# 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 .shout[preferences]
    - WTF does that mean?
]

---

# Preferences

---

# Preferences I

- Which bundles of `$(x, y)$` are **preferred** over others?

`$$\text{Bundle }a=\begin{pmatrix}
			4 \\
			12\\
			\end{pmatrix}
\text{ or Bundle } b=\begin{pmatrix}
			6 \\
			12\\\end{pmatrix}$$`
]

]

---

# Preferences II

- We will allow **three possible answers**:

]

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# Preferences II

- We will allow **three possible answers**:

1. .blue[`\$a \succ b\$`: Strictly prefer `\$a\$` over `\$b\$`]
]

]

---

# Preferences II

- We will allow **three possible answers**:

1. .blue[`\$a \succ b\$`: Strictly prefer `\$a\$` over `\$b\$`]

2. .blue[`\$a \prec b\$`: Strictly prefer `\$b\$` over `\$a\$`]

]

---

# Preferences II

- We will allow **three possible answers**:.red[1]

1. .blue[`\$a \succ b\$`: Strictly prefer `\$a\$` over `\$b\$`]

2. .blue[`\$a \prec b\$`: Strictly prefer `\$b\$` over `\$a\$`]

3. .blue[`\$a \sim b\$`: Indifferent between `\$a\$` and `\$b\$`]
]

- .shout[*Preferences* are a list of all such comparisons between all bundles]
]

.footnote[.red[1] Note you can allow for "weak" preferences: `\$(a \succeq b)\$`, i.e. "`\$a\$` is at least as good as `\$b\$`." We use "strict" preferences to keep it simple.]

---

# 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:
`$$a \succeq a$$`

]

---

# Assumptions About Preferences

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

1. *Reflexivity* : any bundle is at least as preferred as itself:
`$$a \succeq a$$`

2. *Completeness* : any two bundles can be ranked:
`$$a \succeq b \text{ or } a \preceq b \text{ or } a \sim b \; \forall (a,b) \in \mathbb{R}^+$$`

]

---

# Assumptions About Preferences

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

1. *Reflexivity* : any bundle is at least as preferred as itself:
`$$a \succeq a$$`

2. *Completeness* : any two bundles can be ranked:
`$$a \succeq b \text{ or } a \preceq b \text{ or } a \sim b \; \forall (a,b) \in \mathbb{R}^+$$`

3. *Transitivity* : rankings are logically consistent:
  - If `$a \succ b$` and `$b \succ c$`, then `$a \succ c$`

- Are these good assumptions? As usual in economics: *very very often yes, sometimes no!*

]

---

---

# Mapping Preferences Graphically I

.pull-left[
- 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 .shout["indifference curves"] to link bundles of *equal preference*

]

---

# Mapping Preferences Graphically III

3-D "Mount Utility"
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]
]
.pull-right[
.center[
2-D Indifference Curve Contours

]
]
---

# Indifference Curves: Example

.content-box-green[
.green[**Example**]: Suppose you are hunting for an apartment. You value *both* the size of the apartment and the number of friends that live nearby. 
]
]
.pull-right[

]

---

# Indifference Curves: Example

.content-box-green[
.green[**Example**]: Suppose you are hunting for an apartment. You value *both* the size of the apartment and the number of friends that live nearby.

- Apartment `$A$` has 1 friend nearby and is 1,200 `$ft^2$`
]
]
.pull-right[

]

---

# Indifference Curves: Example

.content-box-green[
.green[**Example**]: Suppose you are hunting for an apartment. You value *both* the size of the apartment and the number of friends that live nearby.

- Apartment `$A$` has 1 friend nearby and is 1,200 `$ft^2$`
    - Apartments that are larger and/or have more friends `$\succ A$`
]
]
.pull-right[

]

---

# Indifference Curves: Example

.content-box-green[
.green[**Example**]: Suppose you are hunting for an apartment. You value *both* the size of the apartment and the number of friends that live nearby.

- Apartment `$A$` has 1 friend nearby and is 1,200 `$ft^2$`
    - Apartments that are larger and/or have more friends `$\succ A$`
    - Apartments that are smaller and/or have fewer friends `$\prec A$`
]
]
.pull-right[

]

---

# Indifference Curves: Example

.content-box-green[
.green[**Example**]: Suppose you are hunting for an apartment. You value *both* the size of the apartment and the number of friends that live nearby.

- Apartment `$A$` has 1 friend nearby and is 1,200 `$ft^2$`

- Apartment `$B$` has *more* friends but *less* `$ft^2$`

]
]
.pull-right[

]

---

# Indifference Curves: Example

.content-box-green[
.green[**Example**]: Suppose you are hunting for an apartment. You value *both* the size of the apartment and the number of friends that live nearby.

- Apartment `$A$` has 1 friend nearby and is 1,200 `$ft^2$`

- Apartment `$B$` has *more* friends but *less* `$ft^2$`

- Apartment `$C$` has *still more* friends but *less* `$ft^2$`

]
]
.pull-right[

]

---

# Indifference Curves: Example

.content-box-green[
.green[**Example**]: Suppose you are hunting for an apartment. You value *both* the size of the apartment and the number of friends that live nearby.

- Apartment `$A$` has 1 friend nearby and is 1,200 `$ft^2$`

- Apartment `$B$` has *more* friends but *less* `$ft^2$`

- Apartment `$C$` has *still more* friends but *less* `$ft^2$`

- If `$A \sim B \sim C$`, these apartments are on the same .shout[indifference curve]
]
]
.pull-right[

]

---

# Indifference Curves: Example

.content-box-green[
.green[**Example**]: Suppose you are hunting for an apartment. You value *both* the size of the apartment and the number of friends that live nearby.

- .blue[Indifferent] between all apartments on the same curve
]
]
.pull-right[

]

---

# Indifference Curves: Example

.content-box-green[
.green[**Example**]: Suppose you are hunting for an apartment. You value *both* the size of the apartment and the number of friends that live nearby.

- .blue[Indifferent] between all apts on the same curve

- Apts above the curve are .green[preferred over] apts on the curve
  - `$D \succ A \sim B \sim C$`
  - On a .green[higher curve]
]
]
.pull-right[

]

---

# Indifference Curves: Example

.content-box-green[
.green[**Example**]: Suppose you are hunting for an apartment. You value *both* the size of the apartment and the number of friends that live nearby.

- .blue[Indifferent] between all apts on the same curve

- Apts *above* curve are .green[preferred over] apts *on* curve
  - `$D \succ A \sim B \sim C$`
  - Actually on a .green[higher curve]
]
]
.pull-right[

]

---

# Indifference Curves: Example

.content-box-green[
.green[**Example**]: Suppose you are hunting for an apartment. You value *both* the size of the apartment and the number of friends that live nearby.

- .blue[Indifferent] between all apts on the same curve

- Apts *above* curve are .green[preferred over] apts *on* curve
  - `$D \succ A \sim B \sim C$`
  - Actually on a .green[higher curve]
  
- Apts *below* curve are .red[preferred less] than apts *on* curve
  - `$E \prec A \sim B \sim C$`
  - Actually on a .red[lower curve]
]
]
.pull-right[

]

---

# Assumptions About Indifference Curves

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

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

2. .whisper[Monotonic]: "more is preferred to less"

3. .whisper[Convex]: "averages are preferred to extremes"

4. .whisper[Indifference curves can never cross]: preferences are transitive

]

---

# Assumption 1: We Can Always Draw Them

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

]

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# Assumption 1: We Can Always Draw Them

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

]

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# Assumption 2: Monotonicity

- For any bundle `$b$` with more of at least one good than bundle `$a \implies a \prec b$`

- Moves to NE always preferable

- Moves to SW always unpreferable

]

]

---

# Assumption 2: Monotonicity - Application

---

# Assumption 3: Convexity

- .whisper[Convex]: "averages are preferred to extremes"

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

- Any apt that provides a "balance" of the two desired goods (e.g. `$C) \prec$` "unbalanced" `$(A$` or `$B)$`

- Observation: people prefer variety in consumption

- Mathematically:.red[1] **convex** `$\implies$` a line connecting two points on function lies *above* the function

]
.pull-right[

]

---

# Assumption 3: Convexity - Application

.context-box-blue[
.green[**Application**]: What do non-convex (e.g. concave) indifference curves look like? What kinds of preferences does this imply?
]

---

# Assumption 4: Curves Never Cross

- .whisper[Indifference curves can never cross]: preferences are transitive

- Suppose two curves crossed:

- .blue[`\$A \sim B\$`]

- .orange[`\$B \sim C\$`]

- But .orange[`\$C\$`] `$\succ$` .blue[`\$B\$`]!

- Preferences are not transitive!

]
.pull-right[

]

---

---

# Marginal Rate of Substitution I

.pull-left[
- If I take away one friend nearby, how many more `$ft^2$` would you need to keep you indifferent?

- .shout[Marginal Rate of Substitution (MRS)]: rate at which you trade off one good for the other and *stay satisfied* (remain indifferent)

- Think of this as the **opportunity cost**: # of units of `$y$` you need to give up to acquire 1 more `$x$`

- Recall, **budget constraint** measures *market*-valued trade off between `$x$` and `$y$`

- **MRS** measures your *personal* evaluation of `$x$` vs. `$y$` based on your preferences
]

---

# Marginal Rate of Substitution I

.pull-left[
- If I take away one friend nearby, how many more `$ft^2$` would you need to keep you indifferent?

- .shout[Marginal Rate of Substitution (MRS)]: rate at which you trade off one good for the other and *stay satisfied* (remain indifferent)

- Think of this as the **opportunity cost**: # of units of `$y$` you need to give up to acquire 1 more `$x$`

- Recall, **budget constraint** measures *market*-valued trade off between `$x$` and `$y$`

- **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
`$$MRS_{x,y}=-\frac{\Delta y}{\Delta x} = \frac{rise}{run}$$`

- Amount of `$y$` given up for 1 more `$x$`

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

]
.pull-right[

]

---

---

# So Where are the Numbers?

- Utility thought to be lurking in people's brains
    - Could be understood from first principles: calories, water, warmth, etc
    
- Obvious problems
]

.pull-right[
.center[
![:scale 100%](https://www.dropbox.com/s/j0kwzqb2ptusckt/madlaboratory.jpg?raw=1)
]
]

.footnote[.red[1] "Neuroeconomics" & cognitive scientists are re-attempting a scientific approach to measure utility]

---

# Utility Functions?

- 20th century innovation: .shout[preferences] as the objects of maximization

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

- .whisper[Principle of Revealed Preference]: if `$x$` and `$y$` are both feasible, and if `$x$` is chosen over `$y$`, then the person must (weakly) prefer `$x \succeq y$`
	
- Flawless? Of course not. But extremely useful! 
]

---

# Utility Functions! I

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

- Construct a .shout[utility function] `$u(\cdot)$`.red[1] that *represents* preference relations `$(\succ , \prec , \sim)$`

- Assign utility numbers to bundles, such that, for any bundles `$a$` and `$b$`:
`$$a \succ b \iff u(a)>u(b)$$`
]

.footnote[.red[1] The `\$\cdot\$` 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 `$a \succ b$` 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
]

.footnote[.red[1] The `$\cdot$` is a placeholder for whatever goods we are considering (e.g. `$x$`, `$y$`, `$burritos$`, `$lattes$`, etc)]

---

# Utility Functions, Pural I

.pull-left[
- Imagine three alternative bundles of `$(x, y)$`:
`$$\begin{aligned}
a&=(1,2)\\
b&=(2,2)\\
c&=(4,3)\\
\end{aligned}$$`

]

---

# Utility Functions, Pural I

.pull-left[
- Imagine three alternative bundles of `$(x, y)$`:
`$$\begin{aligned}
a&=(1,2)\\
b&=(2,2)\\
c&=(4,3)\\
\end{aligned}$$`

- Create a utility function `$u(\cdot)$` that assigns each bundle a utility level of

| `$u(\cdot)$` |
|------------|
| `$u(a)=1$`   | 
| `$u(b)=2$`   |
| `$u(c)=3$`   |

- Does it mean that bundle `$c$` is 3 times the utility of `$a$`?

]

---

# Utility Functions, Pural I

.pull-left[
- Imagine three alternative bundles of `$(x, y)$`:
`$$\begin{aligned}
a&=(1,2)\\
b&=(2,2)\\
c&=(4,3)\\
\end{aligned}$$`

- Create a utility function `$u(\cdot)$` that assigns each bundle a utility level of

| `$u(\cdot)$` |
|------------|
| `$u(a)=1$`   | 
| `$u(b)=2$`   |
| `$u(c)=3$`   |

- Does it mean that bundle `$c$` is 3 times the utility of `$a$`?

]

---

# Utility Functions, Pural II

.pull-left[
- Imagine three alternative bundles of `$(x, y)$`:
`$$\begin{aligned}
a&=(1,2)\\
b&=(2,2)\\
c&=(4,3)\\
\end{aligned}$$`

- Now consider `$u(\cdot)$` and a *second* utility function `$v(\cdot)$`:

| `$u(\cdot)$` | `$v(\cdot)$` |
|------------|------------|
| `$u(a)=1$`   | `$v(a)=3$`   |
| `$u(b)=2$`   | `$v(b)=5$`   |
| `$u(c)=3$`   | `$v(c)=7$`   |

]

---

# Utility Functions, Pural III

- Only the preference *ordering** of `$a, b, c$` matters!

- Both are valid:.red[1]
 - `$u(c)>u(b)>u(a)$` 
 - `$w(c)>w(b)>w(a)$`

- Because `$c \succ b \succ a$`

]

.footnote[.red[1] See the Mathematical Appendix in [Today's Class Page](http://microf19.classes.ryansafner.com/class/04-class) for why.]