1.4 — Preferences and Utility

# 1.4 — Preferences and Utility
## ECON 306 • Microeconomic Analysis • Fall 2020
### Ryan Safner<br> Assistant Professor of Economics <br> <a href="mailto:safner@hood.edu"><i class="fa fa-paper-plane fa-fw"></i>safner@hood.edu</a> <br> <a href="https://github.com/ryansafner/microF20"><i class="fa fa-github fa-fw"></i>ryansafner/microF20</a><br> <a href="https://microF20.classes.ryansafner.com"> <i class="fa fa-globe fa-fw"></i>microF20.classes.ryansafner.com</a><br>

---

# Outline

### [Preferences](#5)
### [Indifference Curves](#8)
### [Marginal Rate of Substitution](#23)
### [Utility](#27)
### [Marginal Utility](#42)
### [MRS and Preferences](#42)

---

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

---

# Preferences

---

# Preferences I

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

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

]

---

# Preferences II

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

]

---

# 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**:

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\$`]
]
]

---

# 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\$`]

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

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

See appendix in [today's class page](http://microf20.classes.ryansafner.com/class/1.4-class) for more.
]

---

---

# 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 .hi["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[

]

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# 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.

- Apt. *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.

- Apt. *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.

- Apt. *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

- Apt. *A* has 1 friend nearby and is 1,200 `$ft^2$`

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

]
]
.pull-right[

]

---

# Indifference Curves: Example

- Apt. *A* has 1 friend nearby and is 1,200 `$ft^2$`

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

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

]
]
.pull-right[

]

---

# Indifference Curves: Example

- Apt. *A* has 1 friend nearby and is 1,200 `$ft^2$`

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

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

- If `$A \sim B \sim C$`, on same .hi[indifference curve]
]
]
.pull-right[

]

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# Indifference Curves: Example

- .hi-blue[Indifferent] between all apartments on the **same** curve
]

]

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# Indifference Curves: Example

- .hi-blue[Indifferent] between all apartments on the **same** curve

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

]

---

# Indifference Curves: Example

- .hi-blue[Indifferent] between all apartments on the **same** curve

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

]

---

# Curves Never Cross!

- .hi-purple[Indifference curves can never cross]: preferences are transitive
  - If I prefer `$A \succ B$`, and `$B \succ C$`, I must prefer `$A \succ C$`

]
.pull-right[

]

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# Curves Never Cross!

- .hi-purple[Indifference curves can never cross]: preferences are transitive
  - If I prefer `$A \succ B$`, and `$B \succ C$`, I must prefer `$A \succ C$`

- Suppose two curves crossed:
  - .blue[`\$A \sim B\$`]
  - .orange[`\$B \sim C\$`]
  - But .orange[`\$C\$`] `$\succ$` .blue[`\$B\$`]!
  - Preferences are not transitive!

]
.pull-right[

]

---

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# 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**?

]

---

# 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**?

- .hi[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 `$x$`

]

---

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

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

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

]
.pull-right[

]

---

---

# So Where are the Numbers?

.pull-left[
- Long ago (1890s), utility considered a real, measurable, cardinal scale<sup>.hi[†]</sup>

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

.footnote[<sup>.hi[†]</sup> "Neuroeconomics" & cognitive scientists are re-attempting a scientific approach to measure utility]

---

# Utility Functions?

- 20<sup>th</sup> century innovation: .hi[preferences] as the objects of maximization

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

- .hi-purple[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! 
]

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# Utility Functions! I

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

- Construct a .hi[utility function] `$u(\cdot)$`<sup>.hi[†]</sup> 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[<sup>.hi[†]</sup> 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**:

- .hi-purple["Maximizing preferences"]: choosing `$a$` such that `$a \succ b$` for all available `$b$`

- .hi-purple["Maximizing utility"]: choosing `$a$` such that `$u(a) > u(b)$` for all available `$b$`

- Identical if they contain the same information
]

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# Utility Functions, Pural I

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

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

]

---

# Utility Functions, Pural I

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

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

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

]

---

# Utility Functions, Pural II

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

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

]

---

# Utility Functions, Pural III

.pull-left[
- Utility numbers have an .hi-purple[ordinal] meaning only, **not cardinal**
  - .hi-purple[Only the .ul[ordering] `\$c \succ b \succ a\$` matters!]

- Both are valid:<sup>.hi[†]</sup>
    - `$u(c)>u(b)>u(a)$` 
    - `$v(c)>v(b)>v(a)$`

]

.footnote[<sup>.hi[†]</sup> See the Mathematical Appendix in [Today's Class Page](http://microf20.classes.ryansafner.com/class/1.4-class) for why.]

---

# Utility Functions and Indifference Curves I

.pull-left[
- Two tools to represent preferences: .hi[indifference curves] and .hi[utility functions]

- Indifference curve: all **equally preferred** bundles `$\iff$` **same utility level**

- Each indifference curve represents one level (or contour) of utility surface (function)

]

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# Utility Functions and Indifference Curves II

3-D Utility Function: `$u(x,y)=\sqrt{xy}$`
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2-D Indifference Curve Contours: `$y=\frac{u^2}{x}$`

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# MRS and Marginal Utility I

- Recall: .hi[marginal rate of substitution `\$MRS_{x,y}\$`] is slope of the indifference curve
  - Amount of `$y$` given up for 1 more `$x$`

- How to calculate MRS?
  - Recall it changes (not a straight line)!
  - We can calculate it using something from the **utility function**

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# MRS and Marginal Utility II

- .hi[Marginal utility]: change in utility from a marginal increase in consumption

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# MRS and Marginal Utility II

- .hi[Marginal utility]: change in utility from a marginal increase in consumption

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.green[**Marginal utility of `\$x\$`**]: `$MU_x = \frac{\Delta u(x,y)}{\Delta x}$`
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# MRS and Marginal Utility II

- .hi[Marginal utility]: change in utility from a marginal increase in consumption

.content-box-green[
.green[**Marginal utility of `\$x\$`**]: `$MU_x = \frac{\Delta u(x,y)}{\Delta x}$`
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.content-box-green[
.green[**Marginal utility of `\$y\$`**]: `$MU_y = \frac{\Delta u(x,y)}{\Delta y}$`
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# MRS and Marginal Utility II

- .hi[Marginal utility]: change in utility from a marginal increase in consumption

- Math (calculus): "*marginal*" means "*derivative with respect to*"
  - I will always derive marginal utility functions for you

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# MRS and Marginal Utility: Example

`$$u(x,y) = x^2+y^3$$`

- Marginal utility of x: `$\quad MU_x = 2x$`
- Marginal utlity of y: `$\quad MU_y = 3y^2$`

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# MRS Equation and Marginal Utility

- Relationship between `$MU$` and `$MRS$`:

`$$\underbrace{\frac{\Delta y}{\Delta x}}_{MRS} = -\frac{MU_{x}}{MU_{y}}$$`

- See proof in [today's class notes](/class/1.4-class)

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# MRS and Preferences: Goods, Bads, Neutrals

- More precise ways to classify objects:

- A .hi[good] enters utility function positively
    - `$\uparrow$` good `$\implies$` `$\uparrow$` utility
    - Willing to pay (give up other goods) to *acquire more* (monotonic)
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# MRS and Preferences: Goods, Bads, Neutrals

- More precise ways to classify objects:

- A .hi[good] enters utility function positively
    - `$\uparrow$` good `$\implies$` `$\uparrow$` utility
    - Willing to pay (give up other goods) to *acquire more* (monotonic)
    
- A .hi[bad] enters utility function negatively
    - `$\uparrow$` good `$\implies$` `$\downarrow$` utility
    - Willing to pay (give up other goods) to *get rid of*
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# MRS and Preferences: Goods, Bads, Neutrals

- More precise ways to classify objects:

- A .hi[neutral] does not enter utility function at all
    - `$\uparrow, \downarrow$` has no effect on utility
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# MRS and Preferences: Substitutes

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.green[**Example**]: Consider 1-Liter bottles of coke and 2-Liter bottles of coke 
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- Always willing to substitute between Two 1-L bottles for One 2-L bottle

- .hi[Perfect substitutes]: goods that can be substituted at same fixed rate and yield same utility

- `\$MRS_{1L,2L}=-0.5\$` (a constant!)

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<img src="1.4-slides_files/figure-html/unnamed-chunk-12-1.png" width="504" style="display: block; margin: auto;" />
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# MRS and Preferences: Complements

- Always consume together in fixed proportions (in this case, 1 for 1)

- .hi[Perfect complements]: goods that can be consumed together in same fixed proportion and yield same utility

- `\$MRS_{H,B}=\$` ?

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<img src="1.4-slides_files/figure-html/unnamed-chunk-13-1.png" width="504" style="display: block; margin: auto;" />
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# Cobb-Douglas Utility Functions

- A very common functional form in economics is .hi[Cobb-Douglas]

`$$u(x,y)=x^ay^b$$`

- Where `$a, b >0$` (and very often `$a+b=1$`)
- Extremely useful, you will see it often!
    - Strictly convex and monotonic indifference curves
    - Other nice properties (we'll see later)
    - See the appendix in [today's class page](http://microf20.classes.ryansafner.com/class/1.4-class)
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# Practice

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.green[**Example**]: Suppose you can consume apples `$(a)$` and broccoli `$(b)$`, and earn utility according to:
`$$u(a,b)=2ab$$`

Where your marginal utilities are:

`$$\begin{align*}
MU_a&=2b\\
MU_b&=2a\\
\end{align*}$$`

1. Put `$a$` on the horizontal axis and `$b$` on the vertical axis. Write an equation for `$MRS_{a,b}$`.

2. Would bundles of `$(1, 4)$` and `$(2, 2)$` be on the same indifference curve?

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