1.3 — Preferences

ECON 306 • Microeconomic Analysis • Fall 2022

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/microF22
microF22.classes.ryansafner.com

Outline

Preferences

Indifference Curves

Marginal Rate of Substitution

Utility

Marginal Utility

Preferences

Preferences I

Which bundles are preferred over others?

Example: Between two bundles of $(x, y)$ :

$a = (4, 12) or b = (6, 12)$

Preferences II

We will allow three possible answers:

Preferences II

We will allow three possible answers:

$a ≻ b$ : (Strictly) prefer $a$ over $b$

Preferences II

We will allow three possible answers:

$a ≻ b$ : (Strictly) prefer $a$ over $b$
$a ≺ b$ : (Strictly) prefer $b$ over $a$

Preferences II

We will allow three possible answers:

$a ≻ b$ : (Strictly) prefer $a$ over $b$
$a ≺ b$ : (Strictly) prefer $b$ over $a$
$a \sim b$ : Indifferent between $a$ and $b$

Preferences II

We will allow three possible answers:

$a ≻ b$ : (Strictly) prefer $a$ over $b$
$a ≺ b$ : (Strictly) prefer $b$ over $a$
$a \sim b$ : Indifferent between $a$ and $b$

Preferences are a list of all such comparisons between all bundles

See appendix in today's class page for more.

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”

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 $f t^{2}$

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 $f t^{2}$
- Apts 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 $f t^{2}$
- Apts that are larger and/or have more friends $≻ A$
- Apts that are smaller and/or have fewer friends $≺ A$

Indifference Curves: Example

Example:

Apt. A has 1 friend nearby and is 1,200 $f t^{2}$
B has more friends but less $f t^{2}$

Indifference Curves: Example

Example:

Apt. A has 1 friend nearby and is 1,200 $f t^{2}$
B has more friends but less $f t^{2}$
C has still more friends but less $f t^{2}$

Indifference Curves: Example

Example:

Apt. A has 1 friend nearby and is 1,200 $f t^{2}$
B has more friends but less $f t^{2}$
C has still more friends but less $f t^{2}$
$A \sim B \sim C$ : 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
- $D ≻ A \sim B \sim C$
- On a higher curve

Indifference Curves: Example

Indifferent between all apartments on the same curve
Apts above curve are preferred over apts on curve
- $D ≻ A \sim B \sim C$
- On a higher curve
Apts below curve are less preferred than apts on curve
- $E ≺ A \sim B \sim C$
- On a lower curve

Curves Never Cross!

Indifference curves can never cross: preferences are transitive
- If I prefer $A ≻ B$ , and $B ≻ C$ , I must prefer $A ≻ C$

Curves Never Cross!

Indifference curves can never cross: preferences are transitive
- If I prefer $A ≻ B$ , and $B ≻ C$ , I must prefer $A ≻ C$
Suppose two curves crossed:
- $A \sim B$
- $B \sim C$
- But $C$ $≻$ $B$ !
- Doesn't make sense (not transitive)!

Marginal Rate of Substitution

Marginal Rate of Substitution I

If I find another apt with 1 fewer friend nearby, how many more $f t^{2}$ would you need to keep you satisfied?

Marginal Rate of Substitution I

If I find another apt with 1 fewer friend nearby, how many more $f t^{2}$ would you need to keep you satisfied?
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 relative value you place on good $x$ :

“I am willing to give up $(M R S)$ units of $y$ to consume 1 more unit of $x$ and stay satisfied.”

Marginal Rate of Substitution II

MRS $=$ slope of the indifference curve

$M R S_{x, y} = - \frac{Δ y}{Δ x} = \frac{r i s e}{r u n}$

Amount of $y$ given up for 1 more $x$
Note: slope (MRS) changes along the curve!

MRS vs. Budget Constraint Slope

Budget constraint (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
Foreshadowing: what if these two rates are different? Are you truly optimizing?

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?

More plausibly infer people's preferences from their actions!
- “Actions speak louder than words”
Principle of Revealed Preference: if a person chooses $x$ over $y$ , and both are affordable, then they must prefer $x ⪰ y$
Flawless? Of course not. But extremely useful approximation!
- People tend not to leave money on the table

Utility Functions!

A utility function $u (\cdot)$ ^† represents preference relations $(≻, ≺, \sim)$
Assign utility numbers to bundles, such that, for any bundles $a$ and $b$ : $a ≻ b ⟺ u (a) > u (b)$

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

Utility Functions, Pural I

Example: Imagine three alternative bundles of $(x, y)$ : $\begin{aligned} a & = (1, 2) \\ b & = (2, 2) \\ c & = (4, 3) \end{aligned}$

Utility Functions, Pural I

Example: Imagine three alternative bundles of $(x, y)$ : $\begin{aligned} a & = (1, 2) \\ b & = (2, 2) \\ c & = (4, 3) \end{aligned}$

Let $u (\cdot)$ assign each bundle a utility of:

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

Utility Functions, Pural I

Example: Imagine three alternative bundles of $(x, y)$ : $\begin{aligned} a & = (1, 2) \\ b & = (2, 2) \\ c & = (4, 3) \end{aligned}$

Let $u (\cdot)$ assign each bundle a utility of:

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

Does this mean that bundle $c$ is 3 times the utility of $a$ ?

Utility Functions, Pural II

Example: Imagine three alternative bundles of $(x, y)$ : $\begin{aligned} a & = (1, 2) \\ b & = (2, 2) \\ c & = (4, 3) \end{aligned}$

Now consider a 2^nd 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

Utility numbers have an ordinal meaning only, not cardinal
Both are valid utility functions:^†
- $u (c) > u (b) > u (a)$ ✅
- $v (c) > v (b) > v (a)$ ✅
- because $c ≻ b ≻ a$
Only the ranking of utility numbers matters!

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

Utility Functions and Indifference Curves I

Two tools to represent preferences: indifference curves and utility functions
Indifference curve: all equally preferred bundles $⟺$ same utility level
Each indifference curve represents one level (or contour) of utility surface (function)

Utility Functions and Indifference Curves II

3-D Utility Function: $u (x, y) = \sqrt{x y}$

2-D Indifference Curve Contours: $y = \frac{u^{2}}{x}$

Marginal Utility

MRS and Marginal Utility I

Recall: marginal rate of substitution $M R S_{x, y}$ 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

MRS and Marginal Utility II

Marginal utility: change in utility from a marginal increase in consumption

MRS and Marginal Utility II

Marginal utility: change in utility from a marginal increase in consumption

Marginal utility of $x$ : $M U_{x} = \frac{Δ u (x, y)}{Δ x}$

MRS and Marginal Utility II

Marginal utility: change in utility from a marginal increase in consumption

Marginal utility of $x$ : $M U_{x} = \frac{Δ u (x, y)}{Δ x}$

Marginal utility of $y$ : $M U_{y} = \frac{Δ u (x, y)}{Δ y}$

MRS and Marginal Utility II

Marginal utility: change in utility from a marginal increase in consumption

Math (calculus): “marginal” $⟺$ “derivative with respect to”

$M U_{x} = \frac{\partial u (x, y)}{\partial x}$

I will always derive marginal utility functions for you

MRS and Marginal Utility: Example

Example: For an example utility function:

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

Marginal utility of $x$ : $M U_{x} = 2 x$
Marginal utility of $y$ : $M U_{y} = 3 y^{2}$

Again, I will always derive marginal utility functions for you

MRS Equation and Marginal Utility

Relationship between $M U$ and $M R S$ :

$\underset{M R S}{\underset{⏟}{\frac{Δ y}{Δ x}}} = - \frac{M U_{x}}{M U_{y}}$

See proof in today's class notes

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

Important Insights About Value

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

We can't measure $M U$ 's, but we can measure $M R S_{x, y}$ and infer the ratio of $M U$ 's!
- Example: if $M R S_{x, y} = 5$ , a unit of good $x$ gives 5 times the marginal utility of good $y$ at the margin

Important Insights About Value

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

Important Insights About Value

Value inherently comes from the fact that we must make 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)

Diminishing Marginal Utility

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 $x$ :
- $↓ M U_{x}$
- $↓ M R S_{x, y}$ : willing to give up fewer units of $y$ for $x$

Special Case: Substitutes

Example: Consider 1-Liter bottles of coke and 2-Liter bottles of coke

Always willing to substitute between Two 1-L bottles for One 2-L bottle
Perfect substitutes: goods that can be substituted at same fixed rate and yield same utility
$M R S_{1 L, 2 L} = - 0.5$ (a constant!)

Special Case: Complements

Example: Consider hot dogs and hot dog buns

Always consume together in fixed proportions (in this case, 1 for 1)
Perfect complements: goods that can be consumed together in same fixed proportion and yield same utility
$M R S_{H, B} =$ ?

Cobb-Douglas Utility Functions

A very common functional form in economics is Cobb-Douglas

$u (x, y) = x^{a} y^{b}$

Extremely useful, you will see it often!
- Lots of nice, useful properties (we'll see later)
- See the appendix in today's class page

Practice

Example: Suppose you can consume apples $(a)$ and broccoli $(b)$ , and earn utility according to:

$\begin{aligned} u (a, b) & = 2 a b \\ M U_{a} & = 2 b \\ M U_{b} & = 2 a \end{aligned}$

Put $a$ on the horizontal axis and $b$ on the vertical axis. Write an equation for $M R S_{a, b}$ .
Would you prefer a bundle of $(1, 4)$ or $(2, 2)$ ?
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?
Repeat question 3, but for when you are consuming 2 of each good.

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1.3 — Preferences

ECON 306 • Microeconomic Analysis • Fall 2022

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/microF22
microF22.classes.ryansafner.com

Outline

Preferences

Indifference Curves

Marginal Rate of Substitution

Utility

Marginal Utility

Preferences

Preferences I

Which bundles are preferred over others?

Example: Between two bundles of $(x, y)$ :

$a = (4, 12) or b = (6, 12)$

Preferences II

We will allow three possible answers:

Preferences II

We will allow three possible answers:

$a ≻ b$ : (Strictly) prefer $a$ over $b$

Preferences II

We will allow three possible answers:

$a ≻ b$ : (Strictly) prefer $a$ over $b$
$a ≺ b$ : (Strictly) prefer $b$ over $a$

Preferences II

We will allow three possible answers:

$a ≻ b$ : (Strictly) prefer $a$ over $b$
$a ≺ b$ : (Strictly) prefer $b$ over $a$
$a \sim b$ : Indifferent between $a$ and $b$

Preferences II

We will allow three possible answers:

$a ≻ b$ : (Strictly) prefer $a$ over $b$
$a ≺ b$ : (Strictly) prefer $b$ over $a$
$a \sim b$ : Indifferent between $a$ and $b$

Preferences are a list of all such comparisons between all bundles

See appendix in today's class page for more.

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”

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 $f t^{2}$

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 $f t^{2}$
- Apts 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 $f t^{2}$
- Apts that are larger and/or have more friends $≻ A$
- Apts that are smaller and/or have fewer friends $≺ A$

Indifference Curves: Example

Example:

Apt. A has 1 friend nearby and is 1,200 $f t^{2}$
B has more friends but less $f t^{2}$

Indifference Curves: Example

Example:

Apt. A has 1 friend nearby and is 1,200 $f t^{2}$
B has more friends but less $f t^{2}$
C has still more friends but less $f t^{2}$

Indifference Curves: Example

Example:

Apt. A has 1 friend nearby and is 1,200 $f t^{2}$
B has more friends but less $f t^{2}$
C has still more friends but less $f t^{2}$
$A \sim B \sim C$ : 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
- $D ≻ A \sim B \sim C$
- On a higher curve

Indifference Curves: Example

Indifferent between all apartments on the same curve
Apts above curve are preferred over apts on curve
- $D ≻ A \sim B \sim C$
- On a higher curve
Apts below curve are less preferred than apts on curve
- $E ≺ A \sim B \sim C$
- On a lower curve

Curves Never Cross!

Indifference curves can never cross: preferences are transitive
- If I prefer $A ≻ B$ , and $B ≻ C$ , I must prefer $A ≻ C$

Curves Never Cross!

Indifference curves can never cross: preferences are transitive
- If I prefer $A ≻ B$ , and $B ≻ C$ , I must prefer $A ≻ C$
Suppose two curves crossed:
- $A \sim B$
- $B \sim C$
- But $C$ $≻$ $B$ !
- Doesn't make sense (not transitive)!

Marginal Rate of Substitution

Marginal Rate of Substitution I

If I find another apt with 1 fewer friend nearby, how many more $f t^{2}$ would you need to keep you satisfied?

Marginal Rate of Substitution I

If I find another apt with 1 fewer friend nearby, how many more $f t^{2}$ would you need to keep you satisfied?
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 relative value you place on good $x$ :

“I am willing to give up $(M R S)$ units of $y$ to consume 1 more unit of $x$ and stay satisfied.”

Marginal Rate of Substitution II

MRS $=$ slope of the indifference curve

$M R S_{x, y} = - \frac{Δ y}{Δ x} = \frac{r i s e}{r u n}$

Amount of $y$ given up for 1 more $x$
Note: slope (MRS) changes along the curve!

MRS vs. Budget Constraint Slope

Budget constraint (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
Foreshadowing: what if these two rates are different? Are you truly optimizing?

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?

More plausibly infer people's preferences from their actions!
- “Actions speak louder than words”
Principle of Revealed Preference: if a person chooses $x$ over $y$ , and both are affordable, then they must prefer $x ⪰ y$
Flawless? Of course not. But extremely useful approximation!
- People tend not to leave money on the table

Utility Functions!

A utility function $u (\cdot)$ ^† represents preference relations $(≻, ≺, \sim)$
Assign utility numbers to bundles, such that, for any bundles $a$ and $b$ : $a ≻ b ⟺ u (a) > u (b)$

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

Utility Functions, Pural I

Example: Imagine three alternative bundles of $(x, y)$ : $\begin{aligned} a & = (1, 2) \\ b & = (2, 2) \\ c & = (4, 3) \end{aligned}$

Utility Functions, Pural I

Example: Imagine three alternative bundles of $(x, y)$ : $\begin{aligned} a & = (1, 2) \\ b & = (2, 2) \\ c & = (4, 3) \end{aligned}$

Let $u (\cdot)$ assign each bundle a utility of:

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

Utility Functions, Pural I

Example: Imagine three alternative bundles of $(x, y)$ : $\begin{aligned} a & = (1, 2) \\ b & = (2, 2) \\ c & = (4, 3) \end{aligned}$

Let $u (\cdot)$ assign each bundle a utility of:

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

Does this mean that bundle $c$ is 3 times the utility of $a$ ?

Utility Functions, Pural II

Example: Imagine three alternative bundles of $(x, y)$ : $\begin{aligned} a & = (1, 2) \\ b & = (2, 2) \\ c & = (4, 3) \end{aligned}$

Now consider a 2^nd 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

Utility numbers have an ordinal meaning only, not cardinal
Both are valid utility functions:^†
- $u (c) > u (b) > u (a)$ ✅
- $v (c) > v (b) > v (a)$ ✅
- because $c ≻ b ≻ a$
Only the ranking of utility numbers matters!

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

Utility Functions and Indifference Curves I

Two tools to represent preferences: indifference curves and utility functions
Indifference curve: all equally preferred bundles $⟺$ same utility level
Each indifference curve represents one level (or contour) of utility surface (function)

Utility Functions and Indifference Curves II

3-D Utility Function: $u (x, y) = \sqrt{x y}$

2-D Indifference Curve Contours: $y = \frac{u^{2}}{x}$

Marginal Utility

MRS and Marginal Utility I

Recall: marginal rate of substitution $M R S_{x, y}$ 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

MRS and Marginal Utility II

Marginal utility: change in utility from a marginal increase in consumption

MRS and Marginal Utility II

Marginal utility: change in utility from a marginal increase in consumption

Marginal utility of $x$ : $M U_{x} = \frac{Δ u (x, y)}{Δ x}$

MRS and Marginal Utility II

Marginal utility: change in utility from a marginal increase in consumption

Marginal utility of $x$ : $M U_{x} = \frac{Δ u (x, y)}{Δ x}$

Marginal utility of $y$ : $M U_{y} = \frac{Δ u (x, y)}{Δ y}$

MRS and Marginal Utility II

Marginal utility: change in utility from a marginal increase in consumption

Math (calculus): “marginal” $⟺$ “derivative with respect to”

$M U_{x} = \frac{\partial u (x, y)}{\partial x}$

I will always derive marginal utility functions for you

MRS and Marginal Utility: Example

Example: For an example utility function:

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

Marginal utility of $x$ : $M U_{x} = 2 x$
Marginal utility of $y$ : $M U_{y} = 3 y^{2}$

Again, I will always derive marginal utility functions for you

MRS Equation and Marginal Utility

Relationship between $M U$ and $M R S$ :

$\underset{M R S}{\underset{⏟}{\frac{Δ y}{Δ x}}} = - \frac{M U_{x}}{M U_{y}}$

See proof in today's class notes

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

Important Insights About Value

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

We can't measure $M U$ 's, but we can measure $M R S_{x, y}$ and infer the ratio of $M U$ 's!
- Example: if $M R S_{x, y} = 5$ , a unit of good $x$ gives 5 times the marginal utility of good $y$ at the margin

Important Insights About Value

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

Important Insights About Value

Value inherently comes from the fact that we must make 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)

Diminishing Marginal Utility

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 $x$ :
- $↓ M U_{x}$
- $↓ M R S_{x, y}$ : willing to give up fewer units of $y$ for $x$

Special Case: Substitutes

Example: Consider 1-Liter bottles of coke and 2-Liter bottles of coke

Always willing to substitute between Two 1-L bottles for One 2-L bottle
Perfect substitutes: goods that can be substituted at same fixed rate and yield same utility
$M R S_{1 L, 2 L} = - 0.5$ (a constant!)

Special Case: Complements

Example: Consider hot dogs and hot dog buns

Always consume together in fixed proportions (in this case, 1 for 1)
Perfect complements: goods that can be consumed together in same fixed proportion and yield same utility
$M R S_{H, B} =$ ?

Cobb-Douglas Utility Functions

A very common functional form in economics is Cobb-Douglas

$u (x, y) = x^{a} y^{b}$

Extremely useful, you will see it often!
- Lots of nice, useful properties (we'll see later)
- See the appendix in today's class page

Practice

Example: Suppose you can consume apples $(a)$ and broccoli $(b)$ , and earn utility according to:

$\begin{aligned} u (a, b) & = 2 a b \\ M U_{a} & = 2 b \\ M U_{b} & = 2 a \end{aligned}$

Put $a$ on the horizontal axis and $b$ on the vertical axis. Write an equation for $M R S_{a, b}$ .
Would you prefer a bundle of $(1, 4)$ or $(2, 2)$ ?
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?
Repeat question 3, but for when you are consuming 2 of each good.

1.3 — Preferences

ECON 306 • Microeconomic Analysis • Fall 2022

Ryan Safner Associate Professor of Economics safner@hood.edu ryansafner/microF22 microF22.classes.ryansafner.com

Outline

Preferences

Preferences I

Preferences II

Preferences II

Preferences II

Preferences II

Preferences II

Indifference Curves

Mapping Preferences Graphically I

Mapping Preferences Graphically II

Mapping Preferences Graphically III

Indifference Curves: Example

Indifference Curves: Example

Indifference Curves: Example

Indifference Curves: Example

Indifference Curves: Example

Indifference Curves: Example

Indifference Curves: Example

Indifference Curves: Example

Indifference Curves: Example

Indifference Curves: Example

Curves Never Cross!

Curves Never Cross!

Marginal Rate of Substitution

Marginal Rate of Substitution I

Marginal Rate of Substitution I

Marginal Rate of Substitution II

Marginal Rate of Substitution II

MRS vs. Budget Constraint Slope

Utility

So Where are the Numbers?

Utility Functions?

Utility Functions!

Utility Functions, Pural I

Utility Functions, Pural I

Utility Functions, Pural I

Utility Functions, Pural II

Utility Functions, Pural III

Utility Functions and Indifference Curves I

Utility Functions and Indifference Curves II

Marginal Utility

MRS and Marginal Utility I

MRS and Marginal Utility II

MRS and Marginal Utility II

MRS and Marginal Utility II

MRS and Marginal Utility II

MRS and Marginal Utility: Example

MRS Equation and Marginal Utility

Important Insights About Value

Important Insights About Value

Important Insights About Value

Diminishing Marginal Utility

Special Case: Substitutes

Special Case: Complements

Cobb-Douglas Utility Functions

Practice

Outline

Help

1.3 — Preferences

1.3 — Preferences

ECON 306 • Microeconomic Analysis • Fall 2022

Ryan Safner Associate Professor of Economics safner@hood.edu ryansafner/microF22 microF22.classes.ryansafner.com

Outline

Preferences

Preferences I

Preferences II

Preferences II

Preferences II

Preferences II

Preferences II

Indifference Curves

Mapping Preferences Graphically I

Mapping Preferences Graphically II

Mapping Preferences Graphically III

Indifference Curves: Example

Indifference Curves: Example

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/microF22
microF22.classes.ryansafner.com

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/microF22
microF22.classes.ryansafner.com