1.3 — Preferences

# 1.3 — Preferences

## ECON 306 • Microeconomic Analysis • Spring 2023

### Ryan Safner<br> Associate 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/microS23"><i class="fa fa-github fa-fw"></i>ryansafner/microS23</a><br> <a href="https://microS23.classes.ryansafner.com"> <i class="fa fa-globe fa-fw"></i>microS23.classes.ryansafner.com</a><br>

---

# Outline

### [Preferences](#3)
### [Indifference Curves](#10)
### [Marginal Rate of Substitution](#26)
### [Utility](#32)
### [Marginal Utility](#43)

---

# Preferences

---

# Preferences I

- Which bundles are **preferred** over others?

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**:] Between two bundles of `$(x,y)$`:

`$$a=(4,12) \text{ or } b=(6,12)$$`
]

]

---

# 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](/resources/appendices/1.3-appendix) 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

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.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

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

]

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# Indifference Curves: Example

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

]

---

# Indifference Curves: Example

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

]

---

# Indifference Curves: Example

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]:
.smaller[
- Apt. *A* has 1 friend nearby and is 1,200 `$ft^2$`

- *B* has *more* friends but *less* `$ft^2$`
]
]
]
.pull-right[

]

---

# Indifference Curves: Example

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]:
.smaller[
- Apt. *A* has 1 friend nearby and is 1,200 `$ft^2$`

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

- *C* has *still more* friends but *less* `$ft^2$`
]
]
]
.pull-right[

]

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# Indifference Curves: Example

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]:

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

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

- `$A \sim B \sim C$`: 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]
]

]

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# Curves Never Cross!

- .hi-purple[Indifference curves can never cross]: preferences are .hi[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 .hi[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\$`]!
  - Doesn't make sense (not transitive)!

]
.pull-right[

]

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# Marginal Rate of Substitution I

.pull-left[
- If I find another apt with *1 fewer friend* nearby, how many *more `$ft^2$`* would you need to keep you *satisfied?*

]

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# Marginal Rate of Substitution I

.pull-left[
- If I find another apt with *1 fewer friend* nearby, how many *more `$ft^2$`* would you need to keep you *satisfied?*

- .hi[Marginal Rate of Substitution (MRS)]: rate at which you trade away one good for more of the other and remain *indifferent*

- Think of this as the .hi-purple[relative value] you place on good `$x$`:

> “I am willing to give up `$(MRS)$` units of `$y$` to consume 1 more unit of `$x$` and stay satisfied.”

]

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# Marginal Rate of Substitution II

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# Marginal Rate of Substitution II

- MRS `$=$` .hi[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[

]

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# MRS vs. Budget Constraint Slope

- [Budget constraint](/content/1.2-content) (slope) from before measured the **market’s** tradeoff between `$x$` and `$y$` based on market prices

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

- [.hi-turquoise[Foreshadowing]](/content/1.4-content): what if these two rates are *different*? Are you truly optimizing?
]

---

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# 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”](https://en.wikipedia.org/wiki/Neuroeconomics) & cognitive scientists are re-attempting a scientific approach to measure utility]

---

# Utility Functions?

- More plausibly .hi-turquoise[infer people's preferences from their actions]!
  - “Actions speak louder than words”

- .hi-purple[Principle of Revealed Preference]: if a person chooses `$x$` over `$y$`, and both are affordable, then they must prefer `$x \succeq y$`
	
- Flawless? Of course not. But extremely useful approximation!
  - People tend not to leave money on the table
]

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# Utility Functions!

- A .hi[utility function] `$u(\cdot)$`<sup>.hi[†]</sup> *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, Pural I

.pull-left[
.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.hi-green[Example]: Imagine three alternative bundles of `$(x, y)$`:
`$$\begin{aligned}
a&=(1,2)\\
b&=(2,2)\\
c&=(4,3)\\
\end{aligned}$$`
]
]

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

]

- .hi-turquoise[Does this mean that bundle `\$c\$` is 3 times the utility of `\$a\$`?]

---

# Utility Functions, Pural II

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

]

---

# Utility Functions, Pural III

- Both are valid utility functions:<sup>.hi[†]</sup>
    - `$u(c)>u(b)>u(a)$` ✅
    - `$v(c)>v(b)>v(a)$` ✅
    - because `$c \succ b \succ a$`

- .hi-purple[Only the .ul[ranking] of utility numbers matters!]

]

.footnote[<sup>.hi[†]</sup> See the Mathematical Appendix in [Today's Class Page](resources/appendices/1.3-appendix/#utility-functions-and-pmts) for why.]

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# 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

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.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

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Marginal utility of `\$x\$`**]: `$MU_x = \frac{\Delta u(x,y)}{\Delta x}$`
]

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.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

.bg-washed-red.b--dark-red.ba.bw2.br3.shadow-5.ph4.mt5[

- .hi-red[Math (calculus)]: “*marginal*” `$\iff$` “*derivative with respect to*”

`$$MU_x = \frac{\partial \, u(x,y)}{\partial \, x}$$`

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- I will always derive marginal utility functions for you

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# MRS and Marginal Utility: Example

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**:] For an example utility function:

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

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

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- Again, I will always derive marginal utility functions for you

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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](/content/1.3-content/#derivation-of-mrs-equation-as-ratio-of-marginal-utilities)

> “I am willing to give up `$\frac{MU_x}{MU_y}$` units of `$y$` to consume 1 more unit of `$x$` and stay satisfied.”

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# Important Insights About Value

> “I am willing to give up `$\frac{MU_x}{MU_y}$` units of `$y$` to consume 1 more unit of `$x$` and stay satisfied.”

- We can't measure `$MU$`'s, but we *can* measure `$MRS_{x,y}$` and infer the **ratio** of `$MU$`'s!
  - .hi-green[Example]: if `$MRS_{x,y} = 5$`, a unit of good `$x$` gives 5 times the marginal utility of good `$y$` at the margin

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# Important Insights About Value

- Value is .hi-purple[subjective]
  - Each of us has our own preferences that determine our ends or objectives
  - Choice is .hi-turquoise[forward looking]: a comparison of your .hi-turquoise[expectations] about opportunities
  
- .hi[Preferences are not comparable across individuals]
  - Only individuals know what they give up at the moment of choice
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# Important Insights About Value

- Value inherently comes from the fact that we must make .hi-purple[tradeoffs]
    - Making one choice means *having to give up* pursuing others!
    - The choice we pursue at the moment must be worth the sacrifice of others! (i.e. highest marginal utility)
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# Diminishing Marginal Utility

.hi-purple[The Law of Diminishing Marginal Utility]: each marginal unit of a good consumed tends to provide less marginal utility than the previous unit, all else equal

- As you consume more `$x$`: 
  - `$\downarrow MU_x$`
  - `$\downarrow MRS_{x,y}$`: willing to give up *fewer* units of `$y$` for `$x$`
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# Special Case: Substitutes

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.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="1.3-slides_files/figure-html/unnamed-chunk-12-1.png" width="504" style="display: block; margin: auto;" />
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# Special Case: Complements

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.green[**Example**]: Consider hot dogs and hot dog buns 
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- 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="1.3-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]

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

- Extremely useful, you will see it often!
    - Lots of nice, useful properties (we'll see later)
    - See the appendix in [today's class page](resources/appendices/1.3-appendix/#cobb-douglas-functions)
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# Practice

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

`$$\begin{align*}
u(a,b)&=2ab\\
MU_a&=2b\\
MU_b&=2a\\
\end{align*}$$`

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

2. Would you prefer a bundle of `$(1, 4)$` or `$(2, 2)$`?

3. Suppose you are currently consuming 1 apple and 4 broccoli.
      a. How many units of broccoli are you willing to give up to eat 1 more apple and remain indifferent?
      b. How much *more* utility would you get if you were to eat 1 more apple?

4. Repeat question 3, but for when you are consuming 2 of each good.
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