2. Preferences and Utility Functions

0:43

spend the base of the first half of the

0:45

course going through.

0:46

And so, we're going to start today with

0:48

the demand curve.

0:49

And the demand curve is going to come

0:51

from how consumers make choices.

0:55

Okay? And that will help us derive the

0:57

demand curve. Then we'll turn next to

0:58

supply curve, which will come from how

1:01

firms make uh production decisions.

1:04

But, let's start with the demand curve.

1:06

And we're going to start by talking

1:07

today about people's preferences

1:09

and then the utility functions.

1:11

Okay?

1:12

So, our model of consumer

1:14

decision-making

1:16

is going to be a model of utility

1:18

maximization.

1:21

That's going to be our fundamental

1:22

Remember, this course is all about

1:23

constrained maximization. Our model

1:25

today is going to be a model of utility

1:26

maximization.

1:28

And this model is going to have two

1:29

components.

1:30

There's going to be consumer

1:31

preferences,

1:34

which is what people want, and there's

1:37

going to be a budget constraint, which

1:39

is what they can afford.

1:43

And we're going to put these two things

1:45

together. We're going to maximize

1:47

people's happiness or their choice or

1:49

their their happiness given their

1:50

preferences subject to the budget

1:53

constraint they face. And that's going

1:55

to be the constrained maximization

1:56

exercise that actually through the magic

1:58

of economics is going to yield the

2:00

demand curve. And it's going to yield a

2:01

very sensible demand curve that you'll

2:02

understand intuitively.

2:05

Now, so what we're going to do is do

2:07

this in three steps.

2:09

Step one over the next two lectures.

2:11

Step one is we'll talk about

2:12

preferences.

2:13

How do we model people's tastes?

2:16

We'll do that today.

2:18

Step two is we'll talk about how we

2:19

translate this to utility function. How

2:22

we mathematically represent people's

2:24

preferences in the utility function.

2:26

We'll do that today as well.

2:28

And then next time, we'll talk about the

2:30

budget constraints that people face.

2:33

So, today we're going to talk about the

2:34

maximand. Next time we'll talk about the

2:36

budget constraint.

2:38

That means today's lecture is quite fun.

2:40

Today's lecture is about unconstrained

2:42

choice. We're not going to worry at all

2:44

about what you can afford, what anything

2:45

costs.

2:47

We'll worry about what things cost.

2:48

We're not going to worry about what you

2:48

can afford. Okay? Today's the lecture

2:51

where you won the lottery.

2:52

Okay? You won the lottery, money is no

2:55

object. How do you think about what you

2:57

want?

2:58

Okay? Next time we'll say, "Well, you

2:59

didn't win the lottery." In fact, as

3:00

we'll learn later in the semester, no

3:02

one wins the lottery. Uh that's an

3:03

incredibly bad deal. Um uh but basically

3:07

next time we'll impose the budget

3:08

constraints. But for today, we're just

3:10

going to ignore that and talk about what

3:13

do you want?

3:14

Okay? And to start this, we're going to

3:17

start with the series of preference

3:20

assumptions.

3:25

A series Remember, as I talked about

3:27

last time, models rely on simplifying

3:30

assumptions. Otherwise, we could never

3:31

write down a model. It would go on

3:32

forever.

3:33

Okay? And the key question is, are those

3:36

simplifying assumptions sensible? Uh do

3:39

they do violence to reality in a way

3:41

which makes you not believe the model?

3:43

Or do are they roughly consistent with

3:44

reality in a way that allows you to go

3:46

on with the model?

3:47

Okay? And we're going to impose three

3:49

preference assumptions, which I hope

3:52

will not violate your sense of

3:53

reasonableness.

3:54

The first is completeness.

4:00

What I mean by that is you have

4:03

preferences over any set of goods you

4:06

might choose from.

4:08

You might be indifferent. You might say,

4:10

"I like A as much as B." But, you can't

4:12

say, "I don't care." or "I don't know."

4:15

You say, "I don't care." That's

4:16

indifference. You can't say, "I don't

4:17

know." You can't literally say, "I don't

4:19

know how I feel about this." Um you

4:21

might uh say you're indifferent to

4:23

things, but you won't say, "I don't know

4:25

uh how I feel about something." That's

4:26

completeness.

4:28

Okay? The second is the assumption we've

4:30

all become familiar with since

4:31

kindergarten math, which is

4:33

transitivity.

4:38

If you prefer A to B and B to C, you

4:40

prefer A to C.

4:41

Okay?

4:42

Uh

4:44

that's that's kind of um

4:46

uh I'm sure that's pretty clear. You've

4:47

done this a lot in other classes.

4:49

So, these two are sort of standard

4:51

assumptions you might make in any math

4:53

class.

4:54

The third assumption is the one where

4:55

the economics comes in,

4:57

which is the assumption of

4:58

non-satiation,

5:02

or the assumption of more is better.

5:08

In this class, we will assume more is

5:12

always better than less.

5:15

Okay? We'll assume more is better than

5:16

less. Now, to be clear, we're not going

5:19

to say that the next unit makes you

5:21

equally happy as the last unit. In fact,

5:23

I'll talk about that in a few minutes.

5:24

We'll in fact assume it makes you less

5:26

happy.

5:27

But, we will say you always want more.

5:29

The face of the chance of more or less,

5:31

you'll always be happier with more.

5:33

Okay? And that's the non-satiation

5:35

assumption.

5:36

Okay? I'll talk about that some during

5:38

the lecture, but that's sort of what's

5:40

going to give our models their power.

5:41

That's the sort of new economics

5:43

assumption that's going to give, beyond

5:45

our typical math assumptions, that's

5:46

going to give our models their power.

5:48

Okay?

5:49

So, that's our assumptions.

5:51

So, armed with those,

5:54

I want to start with the graphical

5:55

representation of preferences.

5:58

I want to graphically represent people's

5:59

preferences. And I'll do so through

6:00

something we call indifference curves.

6:07

Indifference curves.

6:10

Okay?

6:11

These are base Indifference curves are

6:12

basically preference maps.

6:15

Essentially, indifference curves are

6:16

graphical maps of preferences.

6:19

Okay?

6:20

So, for example,

6:23

suppose your parents gave you some money

6:26

at the beginning of the semester, and

6:27

you can spend that money on two things.

6:29

Your parents are rich. They give you

6:29

tons of money. Spend that money on two

6:32

things.

6:33

Buying pizza

6:34

or or um or eating cookies.

6:38

Okay?

6:39

So, consider your Consider preferences

6:42

between pizza and cookies. That's your

6:43

two things you can do. Once again, it's

6:45

constrained model. Obviously, in life

6:46

you can do a million things with your

6:47

money. But, it turns out if we consider

6:50

the contrast between doing two different

6:52

things with your money, you get a rich

6:53

set of intuition that you can apply to a

6:55

much more multi-dimensional decision

6:57

case.

6:57

So, let's start with the two-dimensional

6:59

decision case. You've got your money.

7:00

You're going to have pizza or you're

7:02

going to have cookies.

7:03

Okay? Now, consider three choices. Okay?

7:06

Choice A

7:09

is two pizzas

7:10

and one cookie.

7:13

Choice B

7:15

is one pizza,

7:18

one pizza and two cookies, and choice C

7:22

is two pizzas, two cookies.

7:25

Okay? That's the three packages I want

7:26

to compare.

7:28

And I'm going to assume and I'll

7:30

mathematically rationalize in a few

7:32

minutes. But for now, I'm going to

7:33

assume you are indifferent

7:37

between these two packages.

7:39

I'm going to assume you're equally happy

7:40

with two slices of pizza and one cookie

7:42

or two cookies and one slice of pizza.

7:45

Okay?

7:46

I'm going to assume that. But, I'm also

7:49

going to assume you prefer option C to

7:51

both of these.

7:53

In fact, I'm not I'm going to assume

7:54

that because that is what more is better

7:56

gives you.

7:57

Okay? So, you're indifferent between

7:59

these.

8:00

This indifference doesn't come from any

8:01

property I wrote up there. That's an

8:02

assumption. That's just a I just for

8:04

this case I'm assuming that. This comes

8:06

from the third property I wrote up

8:07

there. You prefer package C cuz more is

8:09

always better than less.

8:11

Okay?

8:12

So, now let's graph your preferences.

8:15

And we do so in figure 2.1

8:18

Okay? In the handout. Um

8:20

Okay. So, um

8:23

here's your indifference curve. So,

8:24

we've graphed on the x-axis your number

8:26

of number of cookies. On the y-axis,

8:28

slices of pizza.

8:30

Okay? Now, you have We've graphed the

8:33

three choices I laid here. Choice A,

8:36

which is two slices of pizza and one

8:37

cookie.

8:39

Choice B, which is two cookies and one

8:40

slice of pizza. And choice C, which is

8:42

two of both.

8:44

And I have drawn on this graph your

8:45

indifference curves. The way your

8:47

indifference curves looks is there's one

8:49

indifference curve between A and B

8:51

because those are the points among which

8:53

you're indifferent.

8:54

So, what an indifference curve

8:56

represents is all combinations of

8:59

consumption among which you are

9:01

indifferent. So, we call it indifference

9:03

curve.

9:03

So, an indifference curve, which will be

9:05

sort of one of the big workhorses of

9:07

this course. An indifference curve

9:09

represents all combinations along which

9:12

you are indifferent. You are indifferent

9:13

between A and B. Therefore, they lie on

9:15

the same curve.

9:17

Okay?

9:19

So, that's sort of our our our

9:21

preference map, our indifference curves.

9:23

And these indifference curves are going

9:24

to have four properties.

9:27

Four properties that you have to that

9:30

follow naturally from this It's really

9:32

three and a half. The third and fourth

9:33

are really pretty much the same, but I

9:35

like to write them out as four.

9:36

Four properties that follow from the

9:38

from these underlying assumptions.

9:41

Property one

9:42

is consumers prefer higher indifference

9:45

curves.

9:46

Consumers prefer

9:49

higher

9:51

indifference curves.

9:53

Okay? And that just follow from more is

9:55

better. That is, an indifference curve

9:56

that's higher goes through the package

9:58

that has at least as much of one thing

10:00

and more of the other thing. Therefore,

10:02

you prefer it.

10:03

Okay? So, as indifference curves shift

10:05

out, people are happier.

10:08

Okay? So, on that higher indifference

10:09

curve

10:11

point C, you are happier than points A

10:13

and B because more is better.

10:15

Okay?

10:17

The second

10:19

is that

10:21

indifference curves

10:23

never cross.

10:27

Indifference curves

10:29

uh never cross.

10:31

Okay, actually that's

10:33

I'm going to that's third actually. I

10:35

want to come to that in order. Second,

10:37

third is indifference curves never

10:38

cross. Second is indifference curves are

10:39

downward sloping.

10:44

Second is indifference curves are

10:45

downward sloping.

10:47

Okay? Indifference curves are downward

10:48

sloping. Let's talk about that first.

10:51

Okay?

10:52

That simply comes from the principle of

10:54

non-satiation.

10:57

So, look at figure 2.2.

10:59

Here's an upward sloping indifference

11:00

curve.

11:03

Okay? Why does that violate the

11:05

principle of non-satiation? Why does

11:06

that violate that? Yeah.

11:07

Well, either if you're either you're

11:09

somehow less happy if you have more

11:11

cookies, like or you're less happy if

11:13

you have more pizza. Yeah, but

11:16

And like there's

11:18

and that violates non-satiation.

11:20

Exactly. So, basically you're

11:22

indifferent on this curve you're

11:23

indifferent between one of each and two

11:24

of each. You can't be indifferent. Two

11:26

of each has got to be better than one of

11:27

each.

11:28

So, an upward sloping indifference curve

11:29

would violate non-satiation.

11:31

So, that's the second property of

11:32

indifference curve.

11:34

The third property of indifference curve

11:35

is indifference curves never cross.

11:37

Okay? We can see that

11:40

in figure 2.3.

11:42

Okay? Someone else tell me why this

11:44

violates the properties I wrote up there

11:45

indifference curves crossing.

11:48

Yeah.

11:49

Because B and C

11:51

What's that? Because B and C are

11:53

strictly better.

11:54

Because the B and C, B is strictly

11:56

better. That's right.

11:58

But but they're they're also but they're

12:00

also both on the same curve as A. So,

12:03

you're saying they're both you're

12:04

indifferent with A for both B and C, but

12:06

you can't be because B is strictly

12:07

better than C. So, it violates

12:09

transitivity.

12:10

Okay? So, the problem with crossing

12:12

indifference curves they violate

12:13

transitivity.

12:17

And then finally, the fourth

12:20

is sort of a cute extra assumption, but

12:22

I think it's important to clarify,

12:24

which is that there is only one

12:27

indifference curve through every

12:29

possible consumption bundle.

12:33

Only one

12:34

IC

12:36

through

12:37

every bundle.

12:42

Okay? You can't have two indifference

12:43

curves going through the same bundle.

12:45

Okay? Uh and that's because of

12:47

completeness. If you have two

12:49

indifference curves going through the

12:50

same bundle, you wouldn't know how you

12:51

felt.

12:52

Okay? So, there can only be one going

12:54

through every bundle cuz you know how

12:55

you feel.

12:56

You may feel indifferent, but you know

12:57

how you feel. You can't say I don't

12:59

know.

13:00

Okay? So, that's sort of a extra

13:01

assumption that sort of completes the

13:03

link to the properties.

13:05

Okay?

13:06

So, that's basically how indifference

13:08

curves work.

13:10

Now,

13:11

I find when I I took this course

13:14

before you were

13:15

God, maybe before your parents were

13:16

born. I don't know. Certainly before you

13:18

guys were born. Okay, when I took this

13:19

course,

13:20

I found this course full of a lot of

13:22

lightbulb moments. That is stuff was

13:23

just sort of confusing then boom an

13:24

example really make it work for me. And

13:26

the example that made indifference

13:27

curves work to me was actually during my

13:29

first year up.

13:31

When my year up was with a grad student

13:33

and that grad student had to decide

13:34

where he was going to accept a job. He

13:35

had a series of job offers and he had to

13:36

decide. And basically said, here's the

13:38

way I'm thinking about it. I'm

13:40

indifferent I've I've indifference map

13:42

and I care about two things. I care

13:44

about school location

13:47

and I care about economics department

13:49

quality.

13:53

Okay? I care about the quality of my

13:54

colleagues and the research that's done

13:55

there and the location. And basically

13:57

he's he had two offers.

13:59

One was from Princeton, which he put up

14:02

here. No offense to New Jerseyans, but

14:03

Princeton as a young single person

14:04

sucks. Okay, fine when you're married

14:06

and have kids, but deadly as a young

14:08

single person. And the other so, that's

14:10

Princeton. Down here was Santa Cruz.

14:13

Okay? Awesome can have the most

14:15

beautiful university in America. Okay?

14:17

But not as good an economics department.

14:19

And he decided he was roughly

14:20

indifferent between the two.

14:23

But he had a third offer

14:25

from the IMF, which is a research

14:27

institution in DC, which as he had a lot

14:29

of good colleagues and DC is way better

14:31

than Princeton, New Jersey even if it's

14:32

not as good as Santa Cruz. So, he

14:34

decided he would take the offer at the

14:35

IMF.

14:37

Okay? Even though the IMF had worse

14:39

colleagues than Princeton

14:41

and worse location than Santa Cruz,

14:43

it was still better in combination the

14:45

two of them given his preferences.

14:47

And that's how he used indifference

14:48

curves to help make his decision.

14:50

Okay?

14:52

So, that's sort of an example of uh of

14:54

applying it. Let's see no offense to the

14:55

New Jerseyans in the room, of which I'm

14:57

one, but believe me you'd rather be in

14:59

Santa Cruz.

15:00

Um okay. Uh

15:02

so, now let's go from preferences

15:05

to utility functions.

15:12

Okay? So, now we're going to move from

15:13

preferences,

15:15

which we've represented graphically,

15:17

to utility functions, which we're going

15:19

to represent mathematically. Remember, I

15:21

want you to understand everything in

15:22

this course at three levels:

15:23

graphically, mathematically, and most

15:25

importantly of all intuitively. Okay?

15:28

So, graphic is indifference curves. Now

15:30

we'll come to the mathematical

15:31

representation, which utility function.

15:34

Okay? And the idea is that every

15:36

individual, all of you in this room,

15:38

have a stable, well-behaved underlying

15:41

mathematical representation of your

15:43

preferences,

15:44

which we call utility function.

15:47

Okay? Now, once again, that's going to

15:49

be very complicated and you have

15:50

preference over lots of different

15:51

things. We're going to make things

15:52

simple by writing out two-dimensional

15:54

representation for now of your

15:56

indifference curve. We're going to say,

15:58

how do we mathematically represent your

16:00

feelings about pizza versus cookies?

16:03

Okay? Imagine that's all you care about

16:05

in the world is pizza and cookies.

16:07

How do we mathematically represent that?

16:09

So, for example, we could write down

16:11

that your utility function is equal to

16:14

the square root of the number of slices

16:15

of pizza times the number of cookies.

16:18

We could write that down. I'm not saying

16:19

that's right. I'm not saying it works

16:21

for anyone in this room or even everyone

16:22

in this room, but that is a possible way

16:25

to represent utility.

16:28

Okay? What this would say, this is

16:30

convenient. We will use we'll end up

16:32

using square root form a lot for utility

16:33

functions, a lot of convenient

16:34

mathematical properties.

16:36

And it happens to jive with our example,

16:39

right? Because in this example you're

16:40

indifferent between two pizza and one

16:42

cookie or one pizza and two cookie.

16:44

They're both square root of two.

16:45

And you prefer two pizza and two

16:47

cookies, that's two.

16:48

Okay? So, this gives you a higher

16:50

utility for two pizza and two cookies

16:53

okay? Um uh

16:55

then uh one pizza

16:57

than one pizza and two cookie or two

16:59

pizza and one cookie.

17:02

So, now the question is what does this

17:03

mean? What is utility?

17:05

Okay? Well, utility doesn't actually

17:08

mean anything. There's not really a

17:10

thing out there called utils.

17:12

Okay?

17:14

In other words, utility is not a

17:15

cardinal concept. It is only an ordinal

17:18

concept.

17:19

You cannot say your utility you are you

17:23

cannot literally say my utility is X%

17:26

higher than your utility, but you can

17:27

rank them.

17:29

So, we're going to assume that utility

17:31

can be ranked to allow you to rank

17:33

choices.

17:34

Even if generally we might slip some and

17:36

sort of pretend utility is cardinal for

17:38

some cute examples, but by and large

17:41

we're going to think of utility as

17:42

purely ordinal. It's just a way to rank

17:44

your choices. It's just when you have a

17:46

set of choices out there over many

17:47

dimensions. Like if your choice in life

17:49

is always over one dimension

17:51

and more was better, it would always be

17:53

easy to rank it, right? You never ever

17:54

problem. Once your choice over more than

17:56

one dimension,

17:58

now if you want to rank them, you need

18:00

some way to combine them. That's what

18:01

this function does. It allows you

18:03

essentially to weight the different

18:05

elements of your consumption bundle so

18:07

you can rank them um

18:09

when it comes time to choose.

18:11

Okay? Now, this is obviously incredibly

18:14

simple,

18:15

but it turns out to be amazingly

18:17

powerful in explaining real world

18:19

behavior.

18:20

Okay?

18:21

And so, what I want to do today is work

18:23

with the underlying mathematics of

18:24

utility

18:25

and then we'll come back we'll see in

18:26

the next few lectures how it could

18:28

actually be used to explain decisions.

18:31

So, a key concept we're going to talk

18:33

about

18:34

in this class is marginal utility.

18:38

Marginal utility is just the derivative

18:39

of the utility function with respect to

18:41

one of the elements. So, the marginal

18:43

utility for cookies of cookies is

18:45

utility of the next cookie

18:48

given how many cookies you've had.

18:50

This class can be very focused on

18:52

marginal decision making.

18:55

In economics, it's all about how you

18:56

think about the next unit. Turns out

18:58

that makes life a ton easier. Turns out

19:00

it's way easier to say, do you want the

19:02

next cookie? Than to say, how many

19:03

cookies do you want?

19:06

Because if you want the next cookie,

19:07

that's sort of a very isolated decision.

19:09

You say, okay, I've had this many

19:10

cookies, do I want the next cookie?

19:12

Whereas before you start eating you say,

19:13

how many cookies do you want? That's

19:14

sort of a harder more global decision.

19:16

So, we're going to focus on the stepwise

19:18

decision making process of do you want

19:20

the next unit, the next cookie or the

19:22

next slice of pizza?

19:24

Okay? And the key feature of utility

19:28

functions we'll work with throughout the

19:29

semester

19:31

is that they will feature diminishing

19:36

diminishing marginal utility.

19:39

Marginal utility will fall as you have

19:41

more of a good.

19:43

The more of a good you've had, the less

19:45

happiness you'll derive from the next

19:47

unit.

19:48

Okay?

19:50

Now, we can see that graphically in

19:52

figure 2-4.

19:55

Figure 2-4 graphs on the x-axis the

19:58

number of cookies holding constant

20:00

pizza. So, let's say you're having two

20:01

pizza slices and you want to say,

20:04

"What's my benefit from the next

20:05

cookie?"

20:07

And on the on the left axis, violating

20:09

what I just said like 15 seconds ago, we

20:11

graph utility.

20:13

Now, once again, the util numbers don't

20:14

mean anything. It's just to give you an

20:16

ordinal sense.

20:18

What you see here is that if you have

20:19

one cookie,

20:20

your utility is 1.4, square root of 2 *

20:23

1.

20:25

If you have two cookies, your utility

20:26

goes up

20:28

to square root of 4, which is 2. You are

20:30

happier with two cookies.

20:32

But you are less happy from the second

20:34

cookie than the first cookie.

20:36

Okay? And you can see that in figure if

20:39

you flip back and forth between 2-4 and

20:40

2-5, you can see that.

20:43

Okay? The first cookie,

20:46

going from zero to one cookie, gave you

20:48

one So, in this case, we're not graphing

20:50

the marginal utility. So, figure 2-4 is

20:53

the level of utility,

20:56

which is not really something you can

20:57

measure, in fact. Figure 2-5 is

20:59

something you can measure, which is

21:00

marginal utility. What's your happiness?

21:02

And we'll talk about measuring this from

21:03

the next cookie. You see, the first

21:05

cookie

21:06

gives you

21:08

um a utility and a utility increment of

21:10

1.4.

21:13

Okay? You go from utility of zero

21:15

to utility of 1.4.

21:17

The next cookie gives you a utility

21:18

increment of 0.59.

21:21

Okay, you go from utility of 1.41

21:23

to utility of 2.

21:25

The next cookie gives you a utility

21:26

increment of 0.45, the square root of 3.

21:28

So, now we flip back to the previous

21:30

page. We're going from the square root

21:32

of 4 to We're going from the square root

21:34

of 4, I'm sorry, to the square root of

21:35

6.

21:36

Square root of 6 is only 0.45 more than

21:39

the square root of 4.

21:40

And so on.

21:42

So, each additional cookie

21:44

makes you less and less happy. It makes

21:46

you happier. It has to cuz more is

21:48

better. But it makes you less and less

21:50

happy.

21:51

Okay? And this makes sense.

21:54

Just think about any decision in life

21:56

starting with nothing of something and

21:58

having the first one. Slice of pizza, a

22:00

cookie, deciding on which movie to go

22:02

to.

22:03

The first movie, the one you want to see

22:04

the most,

22:06

okay, is going to make you happier than

22:08

you want to see the one you want to see

22:09

not quite as much.

22:11

The first cookie when you're hungry will

22:12

make you happier than the second cookie.

22:15

The first slice of pizza will make you

22:16

happier. Now, you may be close to

22:17

indifferent where that second slice of

22:19

pizza makes you almost as happy as the

22:20

first.

22:21

But the first will make you happier.

22:23

Okay, if you think about that's really

22:25

sort of that first step. You were hungry

22:27

and that first one makes you feel

22:27

happier.

22:29

Now, but you got to remember you always

22:32

want more cookies.

22:33

Now, you might say, "Wait a second, this

22:35

is stupid. Okay, once I've had 10

22:36

cookies, I'm going to barf."

22:38

The 11th cookie could actually make me

22:39

worse off cuz I don't like barfing.

22:42

But

22:43

in economics, we have to remember you

22:45

don't have to eat the 11th cookie. You

22:47

could give it away.

22:49

So, if I say you want to buy the 11th

22:50

cookie, you could save it for later. You

22:52

could give it to a friend.

22:53

So, you always want it. In the worst

22:56

case, you throw it out.

22:57

It can't make you worse off.

22:59

It can only make you better off.

23:01

And that's what where our sort of more

23:03

is better assumption comes from.

23:04

Obviously, in the limit, you know, if

23:05

you get a million cookies, your garbage

23:07

can gets full, you have no friends to

23:08

give them to. I understand the limit

23:09

these things fall apart. Okay? But

23:12

that's the basic idea of more is better

23:13

and the basic idea of of diminishing

23:15

marginal utility. Okay, any questions

23:16

about that?

23:17

Yeah.

23:20

Utility function can never be negative

23:22

because we have Well, utility Once

23:25

again, utility is not an ordinal

23:26

concept. You can set up utility

23:28

functions such that the number is

23:29

negative. You can set that up. Okay? The

23:31

marginal utility is always positive. You

23:34

always get some benefit from the next

23:35

unit.

23:37

Utility, once again, the measurement is

23:38

irrelevant. So, it can be negative. You

23:39

can set it up. Yeah, I could write my

23:40

utility function like this, you know,

23:42

something like that. So, it could be

23:43

negative. That's just a sort of scaling

23:45

factor. But marginal utility is always

23:47

positive. You're always happier or it's

23:49

not it's not negative. You're always

23:51

happier at least indifferent to getting

23:53

the next unit.

23:54

Yeah. So, when you're looking at 2-5,

23:56

you can't look at the function of the

23:58

pizza and the marginal utility, so it's

23:59

going to go down.

24:02

Uh I'm sorry. You look Figure 2-5, no,

24:04

the marginal utility is going to go

24:05

down.

24:06

Each fraction of cookie, the marginal

24:07

utility Marginal utility is always

24:09

diminishing. So, if you start with zero,

24:11

then you can get half of the pizza in

24:12

this graph.

24:14

Well, it's really hard to do it from

24:15

zero. That's really tricky. It's sort of

24:17

much easier to start from one.

24:19

So, corner solutions, we'll talk a lot

24:20

about corner solutions in this class.

24:21

They get ugly. Think of it starting from

24:23

one. Starting from that first cookie,

24:25

every fraction of a cookie makes you

24:26

happier but less and less happy with

24:27

each fraction. It's a good question.

24:29

All right. Good questions. All right.

24:31

So, now let's let's talk about Let's

24:35

flip back from the math to the graphics

24:38

and talk about where indifference curves

24:40

come from. I just drew them out. But in

24:42

fact, indifference curves are the

24:44

graphical representation of what comes

24:46

out of utility function.

24:49

Okay? And indeed, the slope of the

24:51

indifference curve we are going to call

24:54

the marginal rate of substitution.

24:58

The rate essentially at which you're

24:59

willing to substitute

25:02

one good for the other. The rate at

25:04

which you're willing to substitute

25:05

cookies for pizza is your marginal rate

25:09

of substitution.

25:11

And we'll define that as the slope of

25:14

the indifference curve, delta P over

25:16

delta C.

25:18

That is your marginal rate of

25:19

substitution. Literally, the

25:20

indifference curve tells you the rate at

25:22

which you're willing to substitute. You

25:23

just follow along and say, "Look, I'm

25:25

willing to give up um So, in other

25:28

words, if you look at figure 2-6,

25:30

you say, "Look, I'm indifferent between

25:33

point A and point B.

25:35

One cook- one slice of pizza and I'm

25:37

sorry, one cookie and four slices of

25:39

pizza

25:40

is the same to me as two cookies and two

25:42

slices of pizza. Why is it the same?

25:44

Because they both give me utility square

25:45

root of 4, right? So, given this

25:47

mathematical rep- I'm not saying you

25:49

are. I'm saying given this mathematical

25:50

representation,

25:52

okay, you are indifferent between point

25:54

A and point B.

25:55

So, what that says and what's the slope

25:57

of the indifference curve? What's the

25:58

arc slope between point A and point B?

26:01

The slope is -2.

26:04

So, your marginal rate of substitution

26:05

is -2.

26:07

You are indifferent.

26:09

Okay?

26:10

Um you're indifferent between 1-4 and

26:13

2-2. Therefore, you're willing to

26:15

substitute or give away

26:18

two slices of pizza to get one cookie.

26:22

Delta P delta C

26:24

is uh

26:25

is -2.

26:28

Okay? Now, it turns out you can define

26:31

the marginal rate of substitution over

26:32

any segment of indifference curve. And

26:34

what's interesting is it changes. It

26:36

diminishes. Look what happens when we

26:39

move from two pizzas and two cookies

26:42

to from point B to point C.

26:44

Now, the marginal rate of substitution

26:46

is only -1/2.

26:47

Now, I'm only willing to give up one

26:50

slice of pizza to get two cookies.

26:52

What's happening? First, I was willing

26:53

to give up two slices of pizza to get

26:55

one cookie.

26:57

Now, I'm only willing to give up willing

26:58

to give up one slice of pizza to get two

27:00

cookies. What's happening?

27:02

Yeah. You don't want a cookie as much.

27:03

Because of? Diminishing marginal

27:05

utility. Exactly. Diminishing marginal

27:07

utility has caused the marginal rate of

27:10

substitution itself itself to diminish.

27:12

For those of you who are really kind of

27:13

better at math than I am, it turns out

27:15

technically, mathematically, marginal

27:17

utility isn't always diminishing. You

27:19

can draw cases. MRS is always

27:22

diminishing.

27:23

So, you can think of marginal utility as

27:24

always diminishing. It's fine for this

27:25

class. When you get to higher level math

27:27

and economics, you'll see marginal

27:28

utility doesn't have to diminish. MRS

27:30

has to diminish.

27:32

Okay? MRS is always diminishing.

27:35

As you go along the indifference curve,

27:36

that slope is always falling.

27:40

Okay?

27:41

So, basically,

27:43

what we can write now is how the MRS

27:46

relates to utility function,

27:48

our first sort of mind-blowing result,

27:50

is that the MRS is equal to the negative

27:53

of the marginal utility of cookies over

27:56

the marginal utility of pizza.

27:59

That's our first key definition.

28:02

It's equal to the negative of the

28:04

marginal utility of the good on the

28:05

x-axis over the marginal utility of the

28:07

good on the y-axis.

28:10

Okay? Essentially, the marginal rate of

28:13

substitution tells you how your relative

28:16

marginal utilities evolve as you move

28:19

down the indifference curve.

28:22

When you start at point A,

28:25

you have lots of pizza and not a lot of

28:28

cookies.

28:30

When you have lots of pizza,

28:33

your marginal utility is small.

28:36

Here's the key insight. This is the

28:38

thing which, once again, it's a light

28:40

bulb thing. If you get this, it'll make

28:41

your life so much easier. Marginal

28:43

utilities are negative functions of

28:46

quantity.

28:47

The more you have of a thing, the less

28:50

you want the next unit of it.

28:53

That's why, for example, cookies is now

28:54

in the numerator and pizza is in the

28:55

denominator, flipping from this side.

28:58

Okay? The more you have a good, the less

29:00

you want it.

29:02

So, start at point A. You have lots of

29:04

pizza and not a lot of cookies.

29:07

You don't really want more pizza. You

29:09

want more cookies.

29:10

That means the denominator is small.

29:13

The marginal utility of pizza is small.

29:16

You don't really want it.

29:18

But the marginal utility of cookies is

29:19

high. You don't have many of them. So,

29:21

this is a big number.

29:24

Now, let's move to point B.

29:26

And think about your next decision.

29:28

Well, now

29:30

your marginal utility of pizza,

29:33

if you're going to go from two to one

29:34

slice of pizza, now pizza is worth a lot

29:35

more than cookies. So, now it gets

29:37

smaller.

29:38

So, essentially, as you move along that

29:40

indifference curve, because of this you

29:43

want because diminishing marginal

29:44

utility,

29:45

it leads this issue of a diminishing

29:47

marginal rate of substitution.

29:49

Okay? So, basically, as you move along

29:52

the indifference curve, you're more and

29:53

more willing to give up

29:55

the good on the x-axis to get the good

29:57

on the y-axis. As you move from the

29:59

upper left to the lower right on that

30:00

indifference map, figure 2.6,

30:03

you're more you're more willing to give

30:04

up

30:06

the good on the on the x-axis to get the

30:08

good on the y-axis.

30:11

And what this implies is that

30:13

indifference curves are con-

30:16

indifference curves

30:17

are uh convex to the origin.

30:21

Indifference curves are convex to the

30:22

origin.

30:23

It's very important. Okay, the

30:26

Let's see. The They are They are not

30:28

concave. They're either convex or

30:29

straight. Let's say they're They're not

30:31

concave to the origin.

30:33

Okay, to be technical.

30:34

Indifference curves can be linear. We'll

30:36

come to that.

30:37

But they can't be concave to the origin.

30:39

Why? Well, let's look at the next

30:40

figure, the last figure, figure 2.7.

30:43

What would happen if indifference curves

30:44

were concave to the origin?

30:47

Then that would say moving from

30:51

one pizza, so now I've drawn a a concave

30:54

indifference curve. And with this

30:55

indifference curve, moving from point A

30:57

to point B leaves you indifferent.

31:00

So, you're happy to give up one slice of

31:02

pizza to get one cookie.

31:04

Starting with four slices of pizza and

31:06

one cookie,

31:07

you were happy to give up one slice of

31:09

pizza to get one cookie.

31:12

Now, starting from two and three, you're

31:15

now willing to give up two slices of

31:16

pizza to get one cookie.

31:19

What does that violate? Why Why does

31:20

that not make sense? Yeah.

31:23

Law of diminishing marginal returns.

31:25

Yeah, it's law of diminishing marginal

31:26

utility. Here, you were you were you

31:28

were only You were happy to have one

31:30

slice of pizza to get one cookie. Now

31:31

you were willing to have two slices of

31:32

pizza to get one cookie, even though you

31:33

have less pizza and more cookies. That

31:35

can't be right. As you have less pizza

31:37

and more cookies, cookies pizza should

31:40

become more valuable, not less valuable.

31:42

And cookies should become less valuable,

31:43

not more valuable.

31:45

So, a concave to the origin indifference

31:47

curve would violate the principle of

31:49

diminishing marginal utility and

31:50

diminishing marginal rate of

31:51

substitution.

31:53

Okay? Yeah. What if it's like trading

31:55

cards?

31:56

Okay. I mean, theory and

31:58

I mean, I think you get more of trading

32:00

cards,

32:01

you you you have fewer cards than you

32:03

want to

32:04

trade that.

32:05

That's very interesting. So, in some

32:06

sense,

32:08

what that is saying is that your utility

32:10

function is really over sets. You're

32:11

saying your utility function is over

32:12

trading cards. It's over sets.

32:15

So, basically, that's what sort of you

32:17

know, what's sort of a bit

32:19

you know, our models are flexible. One

32:22

way to say they're loose, another way to

32:23

say they're flexible. So, one But one of

32:26

the challenges you'll face on this

32:27

course is thinking about what is

32:30

decision set over which I'm writing my

32:31

utility function. You're saying it's

32:32

sets, not trading cards. So, that's why

32:34

it happens.

32:36

Okay? Other questions? Good question.

32:38

Yeah, in the back. What about like

32:39

addictive things where like the more you

32:41

have of it, the more you want to buy?

32:43

Yeah, that's that's a really really good

32:45

question. I spent a lot of my research

32:46

life, actually. I do a lot of I did a

32:48

lot of research for a number of years on

32:50

thinking about how you properly model

32:52

addictive decisions like smoking.

32:54

Addictive decisions like smoking,

32:56

essentially, it really is that your

32:59

utility function itself shifts as you

33:02

get more addictive.

33:03

It's not that your marginal utility, the

33:06

next cigarette is still worth less than

33:07

the first cigarette. It's just that as

33:09

you get more addicted, that first

33:10

cigarette gets worth more and more to

33:12

you.

33:13

So, when you wake up in the morning

33:14

feeling crappy, that first cigarette

33:16

still does more for you than the second

33:17

cigarette. It's just the next day you

33:19

wake up feeling crappier.

33:21

Okay? So, we model addiction as

33:23

something where essentially each day

33:26

cigarettes do less and less for you. You

33:28

get essentially adjusted to new You get

33:29

habituated to higher levels.

33:31

And this is why, you know, I do a lot of

33:33

work, you know, this is why,

33:34

unfortunately, we saw last year the

33:36

number of the highest number of deaths

33:37

from accidental overdose in US history,

33:39

72,000 people died from drug overdoses

33:42

last year, more than ever died in

33:43

traffic accidents in our nation's

33:44

history.

33:45

Okay? Why?

33:47

Because people get habituated to certain

33:49

levels. And they used to be used to

33:51

certain levels. So, people get hooked on

33:52

OxyContin.

33:54

They get habituated to a certain level.

33:55

They maybe switch to heroin. And they're

33:57

habituated to a certain level. And now

33:58

there's this thing called fentanyl,

33:59

which is synthetic opioid brought over

34:01

from China, which is incredibly

34:03

powerful. And dealers are mixing the

34:05

fentanyl in with the heroin.

34:07

And the people shoot up not realizing at

34:09

their habituated level, not realizing

34:11

they have this dangerous substance, and

34:12

they overdose and die.

34:14

And that's because they've got

34:15

habituated to higher levels. They didn't

34:16

realize they're getting a different

34:17

product. So, it's not about not

34:18

diminishing marginal utility. It's about

34:19

different underlying different products.

34:22

All right?

34:23

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