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PROFESSOR: Modeling decision
under uncertainty turns out to
be a critical part of what
we do in economics.
And I'll spend today's
lecture talking
about this set of issues.
And, let me just say, the
uncertainty you face now is
nothing compared to the
uncertainty that you'll face
later in life.
So you have uncertainty now
about whether you should study
for the final, or carry an
umbrella, or go on a date with
this person.
I've got uncertainty about
whether I should refinance my
mortgage, or which college to
send my kid to, or how much
life insurance I should buy.
Uncertainty only get more
and more important as
you move on in life.
This is an important issue.
Now, how do we think
about uncertainty?
Well, the tool that we use to
think about uncertainty is,
once again, to make simplifying
assumptions which
allow us to write down sensible
models, but which
capture the key elements of
what we're thinking about.
And the simplifying assumption
here is we move to the tools
of what we call expected
utility theory.
And so, basically, the way we
think about expected utility
theory is the following.
Imagine that I offered you guys
in this class a choice.
And I'm just going to say right
now, there's no right
answer to this.
But I do want you guys
to answer me.
There's no right answer.
Here's the question.
I'm going to give
you a choice.
I'm going to flip a coin.
I have a coin in pocket, and
I'm going to flip it.
And I'm going to offer
you guys the
ability to make a bet.
If it comes up heads,
you win $125.
If it comes up tails,
you lose $100.
Heads, you win a $125.
Tails, you lose $100.
There's no right answer.
How many would take that bet.
How many people would
not take that bet?
Very good.
That's the typical set of
responses I get to this.
Now, what's interesting
is to think about the
parameters of that bet.
And to think about it, let's
take a step back to something
we've discussed already this
semester, the concept of
expected value.
What's the expected value
of that gamble?
The expected value, if you
remember, is the probability
of each outcome times the
value of that outcome.
That is you remember expected
value, which you defined
before, is the probability that
you lose times the value
if you lose plus the probability
that you win times
the value if you win.
That's the expected
value of a gamble.
So, in this context, the
expected value is there's a
50% probability that
you lose, so 0.5.
And if you lose, you lose minus
$100 plus a 50% value
that you win.
It's flipping a coin
after all.
And if you win, you won $125.
So the expected value of
this gamble is $12.50.
On average, if I did this enough
times, you would win
$12.50 per time.
Statistically, if I did this
enough times, you'd
win $12.50 per time.
So, in other words, we
say that this is
more than a fair bet.
A fair bet is one with an
expected value of 0.
A fair bet has an expected
value of 0.
So a fair bet would be
tails you lose $100,
heads you win $100.
This is a more than fair bet.
There's more than 0
expected value.
Yet, the majority of you
would not be willing
to take this bet.
In fact, the majority
of people would
not take this bet.
Why is that?
Why is it that I've dictated
a bet which has a positive
expected value and yet,
people won't take it.
Yeah.
AUDIENCE: But wouldn't that also
depend on how much money
you have.
PROFESSOR: It will absolutely
depend on how much money you
have.
AUDIENCE: Right.
So if I were a richer person,
then losing $100 isn't as
important to me as the chance
of getting $125.
PROFESSOR: OK.
So flesh that out.
Why is that?
Why is it that basically it
would matter how much wealth
you have. Because no matter how
much wealth you have, this
math is impeachable.
It's always a good bet.
So why is it that your state
without much wealth, your
state as college students
without much wealth, what is
it about you that causes you to
not want to take this bet
that's more than fair.
AUDIENCE: So, basically, for me,
the risk of losing or the
state I will be in after I lose
is much greater, well,
for me, a lot more than what
I would be in if win.
PROFESSOR: Exactly.
And there's two possible
reasons for that.
One we're going to push off to
the very end of the lecture.
The main reason we're going
to focus on is because
individuals do not consider
expected value, they consider
expected utility, and
individuals are risk averse.
Expected utility is going to
differ from expected value
when individuals are
risk averse.
Expecting utility is not going
to be the probability times
the value if you lose.
Expected utility is going to
be the probability that you
lose times the utility if you
lose plus the probability that
you win times the utility
if you win.
And utility is not the same as
value, importantly, because
utility functions exhibit
diminishing marginal utility.
Utility functions
are not linear.
Utility functions
are nonlinear.
And, in particular, there's
diminishing marginal utility.
And with diminishing marginal
utility, you're going to not
want bets where there's the
chance you lose is equal to or
even a bit smaller than the
value that you win.
And the basic point is that the
joy of winning is smaller
than the pain of losing with
diminishing marginal utility.
Yeah.
AUDIENCE: Isn't there also a
statistical side to this then?
Because we don't know how many
times we're going to bet.
It might just be once.
We're a lot more comfortable if,
let's say, use the law of
large numbers and say, OK, it's
going to eventually even
out so we'll win
$12.50 a game.
But for the first, let's say 10
or so games, we might get
really unlucky and flip eight
tails and two heads.
PROFESSOR: But, once again, if
you weren't risk averse, you
wouldn't care about that.
Hold that thought.
I'm going to explain why
that isn't true.
So just hold that thought.
So now let's imagine that your
utility functions are the
typical form we've worked
with before, the typical
diminishing marginal utility
form we've worked with before
where utility is the square
root of consumption.
You're casting your mind back
to consumer theory here.
You're going to have to start
integrating the course now,
both consumer and
producer theory.
So remember we said the typical
diminishing marginal
utility function we worked
with was u equals the
square root of c.
Now, let's say you start with
consumption of $100.
Imagine you consume
your income.
Let's say you have consumption
of $100.
Well, then utility is 10.
If you start with a consumption
of $100, your
utility is 10.
Now let's calculate
the expected
utility of this gamble.
The expected utility of this
gamble is that there's a 50%
chance that you lose.
And, if you lose, what
is your utility?
Well you lose $100.
So consumption goes to 0.
So utility is 0 plus a 50%
chance that you win.
Well, what do you
get if you win.
Well, if you win, you go
from $100 to $225.
So your utility is the square
root of $225, or $15.
Your utility is the square
root of $225 or 15.
It's half chance of having 15.
I'm sorry.
So this is a negative.
Yeah.
So utility, if you take this
gamble, is you end up with a
utility of 7.5.
So utility falls.
You move from a utility of 10
without the gamble to a
utility of 7.5 with
the gamble.
Utility is lower with the
gamble, which is why people
decided they didn't want
to take that gamble.
Utility is lower.
And the reason is because given
a utility function of
this form, you are sadder about
losing than happier
about winning.
To see that, we can see that
graphically in Figure 20-1.
This graph's utility against
wealth-- we don't usually
graph utility, because
it's not cardinal.
Remember it's just ordinal.
But the sort of gives you a
sense of the intuition.
This is a graph of utility
against wealth levels.
So you start at point A. You
start with $100 in wealth,
which is consumption.
and utility of 10.
Now, I give you a choice
of a gamble.
That gamble has a 50% chance of
leaving you at 0 and a 50%
chance a leaving you at point
B. So your utility and
expected value is the midpoint
of that chord that runs from 0
to B or point C. Your expecting
utility is lower
than your initial utility.
Why?
Because utility is concave. You
are made so sad by getting
to 0 that it vastly
overcompensate the happiness
you feel moving to $225 because
of the diminishing
marginal utility.
Because, basically, think
of it this way.
Imagine it's your
actual income.
Let's take the point about the
size of the gamble relative to
income seriously.
Imagine, literally, I was asking
you to gamble your
entire income for the year.
And if you lose, you
starve to death.
And if you win, you get
to eat extra nice.
Well, clearly, the disutility
of starving to death vastly
outweighs the extra utility
to eating well.
So, in that extreme example,
if this was your entire
wealth, you can see why you
would have a situation where
you wouldn't want to
take that gamble.
Because if you lost,
you'd die.
And, basically, risk aversion
arises because, basically,
with diminishing marginal
utility you're
made so much sadder.
That steepness at the bottom,
you get so much sadder as you
get towards 0 that it vastly
overcompensates the flatter
part as you move above
your initial point.
So, as you can see, you are
going to end up not wanting
gambles even if they're fair.
Gambles that are fair, that is
positive expected value, might
still lead to a reduction in
your expected utility.
Indeed, let me go further.
You dislike this gamble so
much that if I said the
following, I as your teacher am
going to force you to take
this gamble-- imagine it's
like 100 years ago where
teachers can beat students
and stuff--
I'm going to force you take this
gamble unless you pay me,
you would actually be
willing to pay me to
avoid taking this gamble.
How much would you pay me?
Imagine utilities in dollar
terms. Imagine we're actually
measuring utility in dollar
terms. How much would you pay
me to avoid taking
this gamble.
If I said you either take the
gamble, or you pay me.
You're starting with
a utility of 100.
Yeah?
AUDIENCE: The difference between
the two utilities.
PROFESSOR: Well, the difference
between the two utilities.
So utility is 100 here.
Here utility is 7.5
squared, so 56.25.
So you would actually
pay me $43.75 to
avoid taking this gamble.
Think about that.
I've offered you a more than
fair bet, a very good bet,
which, on average, will yield
you a positive $12.50.
Yet you will pay me $43.75.
You will pay almost half of your
entire wealth to avoid
taking that gamble.
That's pretty incredible
if you think about it.
I've offered you a more than
fair bet, and yet you will pay
me more than half your wealth,
almost half your wealth, to
avoid taking that bet.
So another way to see
this, let's look at
this another way.
How large would I have to make
the positive payoff for you to
take the bet?
Let's look at it that way.
Right now I said you win
$125 with heads.
How much would you have to win
with heads if you were going
to take that bet?
Yeah.
And tell us how you
figured that out.
AUDIENCE: Because you need to
have at least the same utility
as you had before from the
unexpected utility.
So more than half of his
per year utility
would be 20 if he wins.
20 squared is 400.
[INAUDIBLE  PHRASE].
PROFESSOR: Right.
You'd need to win 300.
Because I'd need to take your
utility to 20 if you win.
Only then would you be willing
to take this gamble.
So another way to say it is
that's how fair a gamble would
need to be, how more than
fair it would need to be
before you take it.
You'd need me to pay off 3:1 on
a 50% chance before you'd
take the bet.
And this is just with a typical
looking utility
function of the kind we worked
earlier in the semester.
You didn't look at this earlier
in the semester and
say, wow, that's a bizarre
utility function.
We got sensible answers on our
problems, and problem sets,
and tests, and things,
examples from
square root of c.
That seemed like a sensible
function.
And yet it yields these
incredibly wild predictions
that you would pay people almost
half of your wealth to
avoid engaging in a more
than fair bet.
And that you would need the odds
to be like 3:1 before you
even consider taking a a bet.
That's the power of uncertainty
and the power of
risk aversion.
Really, risk aversion, it's just
the power of diminishing
marginal utility.
The power of diminishing
marginal utility is so key to
driving our decisions.
It's the fact that that first
pizza means so much more to
you than the fifth pizza, that
you really hate outcomes that
don't let you get
the first pizza.
And, as a result, you will pay
a lot to be forced into a
situation where you don't
get any pizzas.
You'll need to be paid a lot in
the state where you do win
to deal with the state
where you don't.
Questions about that?
Now, we can change the example
in some interesting ways to
understand it.
So let's change the example to
say, instead, let's talk about
some alternatives to this
example and how they affect
our intuition.
First alternative, imagine your
utility function instead
of being square root of c, your
utility function was 0.1
times c, a linear utility
function, not a non-linear
utility function.
We can now say that, in that
case, you actually would take
the gamble.
There's a 0.5% chance of 0.
And I chose 0.1 times c, because
your initial utility
is still 10 then.
I normalized this.
So starting with your bundle of
100 you still start at 10.
It gives the same starting
point as
the square root function.
But now your expected utility
from his gamble is 0.5 times 0
plus 0.5 times if you win 125,
your utility is 12.5.
I'm sorry.
It's 22.5.
So your expected utility is
11.25 which is higher than
your starting utility.
So you would take this gamble.
What's changed?
AUDIENCE: No diminishing
marginal utility.
PROFESSOR: No diminishing
marginal utility because now
we are no longer risk averse.
We are what we call
risk neutral.
A linear utility function
yields risks neutrality.
And once you're risk neutral,
you only care
about expected value.
Risk neutral consumers
would only care
about expected value.
And so a linear utility function
will lead to risk
neutrality since you don't have
diminishing marginal utility.
Then you take any
bet that's fair.
You don't care.
You're indifferent between
winning a dollar and losing a
dollar with this utility
function.
It doesn't matter if
you go up or down.
The joy you get from winning
is the same as the pain you
get from losing.
Whereas with this utility
function, the pain you get
from losing exceeds the
joy from winning.
We can see that graphically in
the next figure, Figure 20-2,
the case of risk neutrality.
Here, you start at point
A. You have 100, and
your utility is 10.
Now, I've offered you a gamble
where there's a 50% chance of
getting 0 and a 50% chance of
getting B. Well, that yields
an outcome of c, which
is a higher utility.
So since your utility is linear,
you're risk neutral,
and you'll take any fair bet.
We can go further.
What if utility, instead, was of
the form u equals c squared
over 1,000?
What if this was your
utility function?
Once again, your initial
utility u of 100 is 10.
It's the same starting point.
But this is a utility function
which now if you do this
gamble, your expected utility is
50% times 0 plus 50% times
$225 squared over 1,000
which is 25.3.
That's a huge increase in
utility from this gamble.
So your expected utility with
the gamble is 25.3.
It's a huge increase
in utility.
And that's because this is an
individual where the shape of
the utility function has change
where they don't have
diminishing marginal
utility, they have
increasing marginal utility.
We've never worked with utility
functions like this before.
These are individuals
we call risk-loving.
That is, they are made happier
by winning $1 than they are
made sadder by losing $1.
It's the opposite of all the
intuition we developed earlier
in this course.
It's a crazy utility function.
But the notion of a risk-loving
utility function
is one where literally $1 that
moves you up makes you happier
than $1 that moves you down
makes you sadder.
You can see that
in Figure 20-3.
Here's a risk-loving
utility function.
The individual starts at point
A. They have a choice of a
gamble where they can have a 50%
chance of landing at 0 and
a 50% chance--
Jessica, that B should
be down at the
intersection of dashed lines--
a 50% chance of landing at B
at the intersection of the
dashed lines.
You take the average of
those two, and it's c.
Their utility is way higher with
the gamble than it was
without the gamble.
In fact.
we can go further.
With a risk-loving person, they
would actually take an
unfair bet.
Consider the following bet.
Tails you lose $100,
heads you win $75.
That's a bet with a negative
expected value.
Neither the risk averse nor the
risk neutral person would
take that bet.
But a risk-loving
person would.
If you work out the math, that
bet gives them a gain in
expected utility.
That is a bet with a negative
expected value that gives them
a gain in expected utility.
Why is that?
Because it's the opposite
of diminishing
marginal utility intuition.
They're made so much happier
by winning that they're
willing to take a bet
even if it's a
negative expected value.
Just like the risk averse person
is made so much sadder
by losing, they won't
take a bet even if
it's more than fair.
So you can actually develop all
the opposite predictions
from a risk-loving person.
They'll even take
unfair gambles.
Now, by the way, I skipped over
your earlier question
about risk neutrality.
With risk neutrality, you see it
doesn't matter if you do it
100 times or one time.
If you're risk neutral, you
should take the bet anytime,
because the expected value
is still positive.
Now, you're thinking about
risk aversion where, in
substance, you're more
confident as
the numbers go up.
But if you're risk neutral,
you'll take it no matter how
many times I offer
you that bet.
So to extend this further, let's
go to a third extension
which will develop this
intuition further.
Now imagine that I offer you
guys a different gamble.
And, once again, I really want
you to answer honestly.
Don't try to game me.
Answer honestly.
Now the gamble is if I flip a
coin, tails you lose $1, heads
you win $1.25.
Now how many of you would
take that gamble.
How many would not
take that gamble?
OK.
I hope you're answering
honestly.
But maybe you're just thinking
ahead and realizing that that
gamble is very different.
And why are people more willing
to take that gamble
than they were willing to take
the previous gamble, the same
risk averse people.
Yeah.
AUDIENCE: The difference
in [INAUDIBLE PHRASE].
PROFESSOR: Exactly.
In particular, the utility
function is locally linear.
Let's go back to Figure 20-1.
As you get closer and closer
to A, you could draw,
essentially, a linear segment.
So for an infinitesimal bet,
utility is linear.
So it's linear at point A.
So for small bets, you
become risk neutral.
Even a risk averse person moves
towards risk neutrality
as the bet is small relative
to their resources.
This was the point that
you were making.
Basically if you're a rich
person, you'd probably be
happy to take the $100
and $125 thing.
I'd be happy to do that.
I'm a rich guy.
I'd be happy to do that.
So, basically, what determines
your willingness to take a bet
is going to be about
what's at stake
relative to your resources.
And what you can see is that
if you solve the math here,
that basically expected utility
even with a square
root of c is positive for
that smaller gamble.
Because as it gets smaller
relative to the $100 you start
with, you become roughly
risk neutral.
And then you'll go ahead
and take the gamble.
So at the end of the day what's
going to determine
whether you're going to take a
gamble is going to be your
level of risk aversion and the
size of the risk you're taking
relative to your resources.
The more risk averse you are,
and the bigger the gamble, the
less likely you are to take it
at a given level of fairness.
Questions about that?
All right.
So now that we all understand
expected utility theory.
Now we're going to go on and
talk about why this matters in
the real world and
how we use it.
And I want to talk, in
particular, about two
applications, insurance
and the lottery.
Let's start by talking
about insurance and
why people have insurance.
Because, in fact, given what
we learned in this lecture,
there would be no reason
for insurance.
This lecture tells us why
people have insurance.
Because there's diminishing
marginal utility, and you're
made so much sadder with a
negative outcome, you're
willing to pay you avoid it.
Remember we talked about that
you would be willing to pay
almost $44 to avoid being
forced to take that bet?
That's what insurance does.
Insurance allows you to
avoid taking gambles.
That's what you can think
of insurance as.
It's a way to avoid
taking a gamble.
You're gambling you're
going to get sick.
You're gambling your house
is going to burn down.
These are gambles you
face that are
forced on you by nature.
What insurance does is
allow you to avoid
taking those gambles.
And just like you'd pay me to
avoid the $100, $125 gamble,
you're paying Aetna to avoid
gambling that you might have
to go to the hospital.
So let's say there's a
25-year-old who is deciding
whether to buy health
insurance.
And let's say they're
25-year-old
guy, totally healthy.
I say guy because there's
no risk they're
going to have a kid.
So he's basically totally
healthy, basically zero chance
they're going to use the
doctor except if they
get hit by a car.
So imagine the situation is that
you've got 25-year-old
with an income of $40,000.
And let's say that there's
a 1% chance that they'll
get hit by a car.
It is Cambridge after all.
So every time you cross the
street, there's a 1% chance
you get hit by a car.
And if you get hit by a car,
you're going to suffer $30,000
in hospital bills.
And let's say your utility
function is square root of c.
So you're a risk averse guy.
So let's say that I then come
to you and say, look, each
year there's an expected
cost to you of getting
hit by car of $300.
How did I calculate that?
Well, every year there's a
1% chance you get hit.
They're independent
draws, let's say.
If you get hit this year,
it doesn't mean
suddenly you're safer.
It's random.
It's just crazy drivers.
So there's a 1% chance you're
going to get hit every year.
And if you get hit, there's a
$30,000 cost. So every year
there's an expected
cost to you--
the opposite of expected value
is expected cost--
of $300.
So let's say I offered to sell
you insurance for $300.
I offered to sell you insurance
in a way where, on
average, if you lived an
infinite number of years, you
would pay out in premiums what
you'd get in benefits.
If you paid $300 a year and
lived forever or lived for
many, many years-- the law of
large numbers enough years--
then basically you would pay out
in premiums what you would
collect in benefits.
You'd get hit once
every 100 years.
And ever 100 years you would
have paid $30,000 in premiums,
and you'd collect $30,000
in benefits.
So that's what we call
actuarially fair insurance.
Actuarially fair insurance is
insurance where the price of
the insurance equals the
probability of the bad outcome
times the cost of
the bad outcome.
That's actuarially fair
insurance where the price you
pay is the probability of the
bad outcome times the cost of
the bad outcome.
That's fair because, over a
large enough population, the
premiums that get paid in will
get paid out in the form of
claims.
Now, let's ask what is your
utility if you do not or do
buy insurance.
So for the first thing you
say if I'm a 25-year-old.
Screw it.
I'm never going to
get hit by a car.
I'm not going to
buy insurance.
What's your utility
with no insurance?
Well, if you have no insurance,
there's a 1%
chance, 0.01, that you'll
lose $30,000.
You'll get hit by a car
and lose $30,000.
Your income is $40,000.
So there's a 1% chance that
you'll end up with a utility,
which is the square
root of 10,000.
And there's a 99% chance you'll
end up with a utility
that's the square
root of 40,000.
You work this out, and the
answer is you get 199.
Utility without insurance is 199
which is pretty close to
utility just if you weren't
going to get hit by the car.
Because it's so rare that
you get hit by the car.
So utility is 199 without
insurance.
Now, let's ask the question, how
much would you be willing
to pay to have insurance?
How do we figure that out?
$300 is the actuarially
fair premium.
But now let's do a different
question.
I'm an insurance company, and
I want to make money.
I don't want to just charge the
actuarially fair premium.
The insurance company
makes no money with
this premium of $300.
So the insurance company
wants to make money.
How would we figure out how much
would you be willing to
pay, this 25-year-old,
be willing
to pay to get insurance?
How do we figure that out?
Yeah.
AUDIENCE: Maybe you could keep
the utility function constant.
PROFESSOR: Keep the utility
value constant.
AUDIENCE: Value, yes.
PROFESSOR: Exactly.
You'd have ask well, how much
would I be willing to pay to
have insurance which would
protect me and leave me at the
same utility level.
Obviously it would have to
be a little bit higher.
But let's just set it equal.
So, in other words, if I bought
insurance, my utility
with insurance, there's a
1% chance that I will
get hit by the car.
In that case, what
happens to me?
Well, if I get hit by the
car, I get $10,000.
I make $40,000.
I lose $30,000.
Let me actually write it out.
If I get hit by the car,
what happens to me?
Well, I make $40,000.
I always make $40,000
each year.
I lose $30,000, because
I get hit by the car.
But then the insurance company
pays me $30,000.
They pay off my debts.
So then I gain $30,000.
So these things cancel.
But I have to pay the insurance
company premium.
So I have to pay
some amount x.
If I don't get hit by the car, I
get my $40,000 income, but I
still have to pay the insurance
company premium.
I have to pay them whether
I get hit or not.
It's insurance.
I pay them either way.
So that's my utility.
So my expected utility
with insurance is the
sum of these two.
And I want to set that
equal to 199.
I want to say what x am I
willing to pay that would
leave me at the same utility as
if I was uninsured as per
the answer here?
Well, it turns out that if you
do, that if you solve this,
you get that x equals 399.
That is you would pay $399
for insurance that has a
value of only $300.
You'd pay $399 for insurance
even though the actuarially
fair price is $300.
You would pay you insurance
company $99 more than they
expect to pay out to you.
Why?
Because you're risk averse.
Because you're made so much
sadder than being left with
$10,000 than you are by
having to pay $300.
If it doesn't work out,
you pay $300.
Who cares?
That's tiny compared
to your income.
But if it does work
out, you're safe
from having to starve.
You pay $400, I'm sorry.
You pay $399.
You're like, look, I'll be
bummed if I have to pay $400.
That's a percent of my
income basically.
That would be a shame to pay
a percent of my income for
something that doesn't happen.
But, boy, would I be happy in
that 1 in 100 chance where I
get hit by a car when
I'm not out $30,000.
So you will pay $399
for insurance
that's only worth $300.
That extra $99 we call
a risk premium.
We call that a risk premium.
The extra $99, we call
a risk premium.
That is the amount that you are
willing to pay above and
beyond the fair price, because
you're risk averse.
And what you should go home and
show yourself using the
same kind of mathematics is
that, for example, the risk
premium will rise the
bigger the loss is.
Hopefully you can see the
intuition on that.
The bigger the loss is for a
given level of income the
bigger the risk is.
Likewise, for a given
loss, the risk
premium falls with income.
So the bigger is the loss of
relative to income the more
risk premium you're
willing to pay.
You should also, obviously, see
that the more risk averse
you are, the bigger premium
you're willing to pay.
A risk neutral person would
not pay a risk premium.
Only a risk averse
person will.
So the more risk averse you are,
the bigger risk premium
you'll pay, and the
bigger the loss is
relative to your income.
These are the same principles
we talked about before.
So the $43.75 we were willing to
pay to avoid that gamble I
was going to force on you, that
was the risk premium.
You were willing to pay $44
to avoid that gamble.
Here, you're willing to pay
$99 to avoid the risk of
ending up in that bad state
where you get hit by the car.
And that's why people
buy insurance.
And that's why insurance
companies make
ungodly amounts of money.
In the US we have a health
insurance industry, for
example, that earns about
$800 billion a year.
Why do they make
all that money?
Because people are risk averse,
and they're willing to
pay to have someone else
bear the risk of
their injury or illness.
Any questions about that?
Now, I don't mean by that to
say, insurance is a bad thing,
and we shouldn't do it.
Risk aversion is the nature
of our utility functions.
We should be willing to
pay a risk premium.
It's just that you need to
understand why, in fact, it
makes sense to have insurance
in that case.
The second application
is the lottery.
The lottery is a total ripoff.
I hope you knew this already.
The expected value
of $1 lottery
ticket is roughly $0.50.
So for every $1 you spend in the
lottery, in expectation,
you get about $0.50 back.
This is an incredibly bad bet,
incredibly unfair, an
incredibly unfair bet.
On average, you lose $0.50
for every $1 you bet.
So, basically, despite that,
lotteries are wildly popular.
They've become a huge source
of revenue for state
governments.
A lot of the money that state
governments now take in is
through state lotteries.
What accounts for the fact that
lotteries are so popular?
Well, there's four different
theories for why lotteries are
so popular.
The first is that people
are risk-loving.
We have it all wrong.
Actually people like taking
risks, and the
lottery feeds that.
This, of course, we can
immediately rule out.
How?
How do we know this is wrong?
That the answer is that people
play the lottery because
they're risk-loving.
How do we know people
aren't risk-loving?
AUDIENCE: The same people don't
take [UNINTELLIGIBLE]
PROFESSOR: And they spend
$800 billion a
year on health insurance.
Basically, as a society, we
spend, in total, about $1.5
trillion a year on insuring
various risks that face us.
We're not risk-loving.
So that's clearly
not the answer.
However, there's
an alternative.
People could basically alternate
between risk-loving
and risk-aversion.
This is a theory due to Milton
Friedman, the famous economist
from Chicago and a co-author
named Savage, the
Friedman-Savage preferences,
where the notion is that
basically people are risk averse
over small gambles but
risk-loving over
large gambles.
So to see that, go last
figure in the graph.
This is sort of a complicated
case.
Basically, the notion is if you
take someone, they have a
utility function which is
initially risk averse and then
becomes risk-loving.
That is in the segment between
W1 and W3, that looks like a
risk averse utility function.
But once you get above W3, it
looks like a risk-loving
utility function.
So the notion is that for things
which can make me very
poor, I'm risk averse.
I want to insure against events
which will leave me in
that bottom segment.
But once I'm going to be
above W3, then great.
I'm happy to take risks.
Then I become risk-loving.
Now, this is a not crazy idea.
Graphically, what I'm showing
you here, is that b* is
utility without the gamble
and b is with.
So you see you're happier
without the gamble when your
income is low.
Once your income is a lot
higher, you're happier with
the gamble at d then you are
without the gamble at d*.
That's not a crazy theory.
The notion is that once I'm
rich enough, I become
risk-loving.
But when I'm poor, I don't
want to take the risks.
The problem is that this is
inconsistent with lottery
behavior in the following
sense.
Most people who play
the lottery don't
play the Mega Millions.
They play tiny scratch
lotteries where you
bet $1 to win $10.
And people spend huge amounts
of money on lotteries with
very, very low payoffs.
That is inconsistent
with this.
Because this would say that
you'd only play lotteries that
have big payoffs.
Lotteries that have small
payoffs, once again, there's
no reason to play that and
still buy insurance.
So if you're buying insurance
against being low income, why
are you playing these small
lotteries that are a ripoff.
Because those small ones
are a ripoff too.
So the existence of the fact
that the most popular
lotteries are actually the
small lotteries is
inconsistent with this
explanation.
Yeah.
AUDIENCE: So I'm confused.
Is it risk-loving on
large gambles?
PROFESSOR: Yeah, risk-loving
on large gambles.
It's not the size
of the gamble.
You're risk-loving on gambles
which leave you in a high
wealth state.
The point is that if
I'm gambling over
winning Mega Millions.
Yeah, I'm a little
risk averse.
But the truth is winning Mega
Millions would make me so
happy that I could move into
the risk-loving part of my
utility function.
But this would not explain why
people ever play something
that pays off $100.
This is a fancy way of the
intuition you probably have.
It's I'd think differently about
something which would
completely change my life and
make me a multi-billionaire,
that's something that would make
me raise me, than the bet
I offered you guys before.
People are systematically taking
terrible bets like the
kind i offered you
guys before.
And that's inconsistent with
these preferences.
The third explanation
is entertainment.
It's that the utility function
has in it the
thrill of the risk.
We only write down utility
functions that are a function
of consumption like how many
pizza and movies you see.
But people have utility
over lots of things.
One thing you may have utility
of the thrill of being able to
scratch the thing off and seeing
if they won or not.
That would actually be
consistent with the fact that
people play a lot of
small lotteries.
If it's a thrill of winning
that matters, if it's the
scratch off thrill that matters,
then the optimal
thing to do, in fact, would be
to not play one Mega Million.
It would be to play lots
of little lotteries.
And that would be consistent
with that behavior.
So one story that is consistent
with what we see is
that people actually view
this as entertainment.
On the other hand, once again,
it's really expensive
entertainment.
Because you're throwing away
$0.50 of every $1.
So you've got to get a lot of
enjoyment out of that scratch
off relative to when you
go to see a movie.
So that's another theory.
I'm going to put this in here.
It sort of inserts in here.
We talked about the fact that
people can't be risk-loving
because they buy insurance.
And this alternating thing
doesn't work, because they
play small lotteries.
But another theory that might
fit here is a theory we call
loss aversion.
This is sort of a different
version of the Friedman-Savage
preferences.
It's that people are, in
general, risk averse.
But, in fact, they're really
risk averse on the downside,
and they don't care so
much on the upside.
So, in other words, the point
is that when I initially
offered you that bet of win
$125, lose $100, part of your
reaction was about the
risk aversion.
But a lot of you are thinking,
I'd be really
bummed if I lost $100.
It's not just that I don't
have it to spare.
It's just like, god, I
would kick myself.
It was one flip of the coin.
How could I possibly have
been so stupid?
Whereas if you won,
you'd be happy.
But then you'd go on
to the next class.
The notion is that basically
it's an extreme version of
risk aversion.
It's not only that you're
risk averse, it go
further than that.
Relative to the starting point,
anything which is a
loss really pisses you off.
So, in fact, even that little
gamble I offered you, win
$1.25 lose $1, you still
might not take.
Some of you still wouldn't
take it.
And the reason you
wouldn't take it
can't be risk aversion.
Because it's just too small
for risk aversion
to plausibly work.
It's that you'll just be bummed
that you did that and
you took that chance.
You'd be made sadder by the
loss than you'd be made
happier by the win.
In that case, that could explain
why people spend a lot
of money to buy insurance.
Because they'll be so bummed
if things go badly.
But they might play the lottery
because, in fact,
around that point, they don't
view the money they're
spending as a loss.
They think of it differently.
They think of the loss of being
my house burned down.
That's a loss.
That would make me really sad.
But the $1 I paid to pay
the lottery, that's
not really a loss.
So I'm risk neutral going
up and really risk
averse going down.
So I'm willing to take gambles
that push me up.
It's sort of like
Friedman-Savage.
I'm willling to take gambles
that push me up, not gambles
that pull me down.
But, once again, that doesn't
really explain the small ones.
That doesn't really explain
the small ones.
That's more the entertainment
theory.
Then finally, the last
theory we have is
that people are stupid.
The lottery is, after all,
its official motto is
a tax on the stupid.
And that's what it is.
It's a tax on the stupid.
Basically many of your public
schools are financed by taxes
paid by stupid people.
It's sort of ironic.
But people just don't know.
You probably all had a vague
sense that the lottery wasn't
a sensible thing to play.
But how many people actually
knew it was that
bad a deal as I said.
That is actually was $0.50
expected payoff.
A few of you knew.
But most of you knewm had a
vague sense it was a bad deal.
You didn't know how
bad a deal it was.
This is sort of hard
to figure out.
Meanwhile, you see on TV that
these guys win these bazillion
dollars, and you get the thrill
of scratching if off.
So, basically, if people are
just stupid, then that could
explain it.
The problem is it matters a
lot for government policy
which of these is right.
Because if A through C is right,
if one through three
are right, then the government
should go
ahead and allow lotteries.
And there's no reason why the
state shouldn't run a lottery.
In fact, let's take the
entertainment theory.
If this is really entertainment,
and the state
can make money off of
my entertainment,
then that's a win-win.
I'm happy, because I'm
playing the lottery.
The state is happy, because
it's financing schools.
That's a win-win.
So if these are right, you're
going to want to encourage
state lotteries.
But if this one's right, we
don't want to have them.
Because, A terrible way to raise
government revenues is
to tax stupid people.
There are much better ways to
raise government revenues.
We'll talk about taxation
in a couple of lectures.
But, clearly, taxing the stupid
is not going to be an
optimal tax.
Yeah.
AUDIENCE: I can maybe sort of
understand why people would
prefer smaller lotteries
over bigger lotteries.
Because they are thinking that
in smaller lotteries, they
have a much bigger
chance of winning
than in bigger lotteries.
So, in that sense, their
expected payoff in terms of
utility or other
[INAUDIBLE PHRASE]
is a lot higher than the antes
in the bigger ones, even
though the bigger ones might end
up being a lot heavier--
PROFESSOR: So that's sort of an
entertainment theory, which
is my utility derives
from the win.
You have a theory in mind my
utility derives from the win.
Because if it's just
about dollars, that
wouldn't explain it.
Because I win so many more from
the big one that it would
compensate from the frequency
at which I'd
win the little one.
But if I actually, in my utility
function, have the joy
of seeing that winning thing,
then that would explain it.
That's an entertainment
theory.
You're saying, in my utility
function, I actually get joy
from scratching off and seeing
that it's a winner, and so
much joy that I'd much rather
take a 10% chance at a small
win than a 1% chance
at a huge win.
Because then, at least, with the
first one, 1 in 10 times I
get that joy of the scratch
off and seeing it's a win.
So that's sort of
an explanation.
And that would say that
lotteries are good.
The other way economists might
think about lotteries is
they're voluntary taxes.
The public doesn't like taxes.
Here's a voluntary tax.
You never hear policy makers
getting up and railing against
a horrible evils
of the lottery.
Sometimes groups do.
Sometimes outside groups
do and stuff.
But politicians don't.
But those same politicians will
go on and on about how
terrible taxes are.
I'm going to cut your taxes.
Taxes are terrible.
Well, the lottery is a voluntary
tax in that sense.
And I might say, look, there's
no reason to oppose it, it's a
voluntary tax.
It's those involuntary
taxes that
cause problems in society.
Well, whether we want to buy
that story or not depends on
how much we think it's being
played because people are
stupid or not.
OK.
Let me stop there.
So that's a great example of
how a little bit of an
extension of our model
can really enrich our
understanding about a lot of
decisions that we make in the
real world.
We'll come back and
talk about another
version like that later.
And that is the case of
thinking about savings
decisions and thinking about
individual decisions on how
much to save and how
much to spend.
