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PROFESSOR: Modeling decision
under uncertainty turns out to

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be a critical part of what
we do in economics.

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And I'll spend today's
lecture talking

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about this set of issues.

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And, let me just say, the
uncertainty you face now is

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nothing compared to the
uncertainty that you'll face

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later in life.

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So you have uncertainty now
about whether you should study

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for the final, or carry an
umbrella, or go on a date with

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this person.

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I've got uncertainty about
whether I should refinance my

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mortgage, or which college to
send my kid to, or how much

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life insurance I should buy.

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Uncertainty only get more
and more important as

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you move on in life.

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This is an important issue.

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Now, how do we think
about uncertainty?

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Well, the tool that we use to
think about uncertainty is,

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once again, to make simplifying
assumptions which

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allow us to write down sensible
models, but which

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capture the key elements of
what we're thinking about.

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And the simplifying assumption
here is we move to the tools

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of what we call expected
utility theory.

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And so, basically, the way we
think about expected utility

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theory is the following.

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Imagine that I offered you guys
in this class a choice.

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And I'm just going to say right
now, there's no right

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answer to this.

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But I do want you guys
to answer me.

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There's no right answer.

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Here's the question.

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I'm going to give
you a choice.

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I'm going to flip a coin.

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I have a coin in pocket, and
I'm going to flip it.

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And I'm going to offer
you guys the

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ability to make a bet.

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If it comes up heads,
you win $125.

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If it comes up tails,
you lose $100.

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Heads, you win a $125.

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Tails, you lose $100.

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There's no right answer.

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How many would take that bet.

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How many people would
not take that bet?

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Very good.

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That's the typical set of
responses I get to this.

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Now, what's interesting
is to think about the

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parameters of that bet.

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And to think about it, let's
take a step back to something

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we've discussed already this
semester, the concept of

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expected value.

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What's the expected value
of that gamble?

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The expected value, if you
remember, is the probability

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of each outcome times the
value of that outcome.

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That is you remember expected
value, which you defined

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before, is the probability that
you lose times the value

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if you lose plus the probability
that you win times

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the value if you win.

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That's the expected
value of a gamble.

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So, in this context, the
expected value is there's a

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50% probability that
you lose, so 0.5.

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And if you lose, you lose minus
$100 plus a 50% value

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that you win.

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It's flipping a coin
after all.

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And if you win, you won $125.

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So the expected value of
this gamble is $12.50.

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On average, if I did this enough
times, you would win

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$12.50 per time.

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Statistically, if I did this
enough times, you'd

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win $12.50 per time.

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So, in other words, we
say that this is

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more than a fair bet.

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A fair bet is one with an
expected value of 0.

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A fair bet has an expected
value of 0.

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So a fair bet would be
tails you lose $100,

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heads you win $100.

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This is a more than fair bet.

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There's more than 0
expected value.

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Yet, the majority of you
would not be willing

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to take this bet.

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In fact, the majority
of people would

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not take this bet.

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Why is that?

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Why is it that I've dictated
a bet which has a positive

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expected value and yet,
people won't take it.

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Yeah.

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AUDIENCE: But wouldn't that also
depend on how much money

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you have.

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PROFESSOR: It will absolutely
depend on how much money you

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have.

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AUDIENCE: Right.

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So if I were a richer person,
then losing $100 isn't as

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important to me as the chance
of getting $125.

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PROFESSOR: OK.

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So flesh that out.

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Why is that?

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Why is it that basically it
would matter how much wealth

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you have. Because no matter how
much wealth you have, this

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math is impeachable.

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It's always a good bet.

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So why is it that your state
without much wealth, your

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state as college students
without much wealth, what is

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it about you that causes you to
not want to take this bet

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that's more than fair.

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AUDIENCE: So, basically, for me,
the risk of losing or the

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state I will be in after I lose
is much greater, well,

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for me, a lot more than what
I would be in if win.

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PROFESSOR: Exactly.

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And there's two possible
reasons for that.

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One we're going to push off to
the very end of the lecture.

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The main reason we're going
to focus on is because

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individuals do not consider
expected value, they consider

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expected utility, and
individuals are risk averse.

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Expected utility is going to
differ from expected value

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when individuals are
risk averse.

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Expecting utility is not going
to be the probability times

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the value if you lose.

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Expected utility is going to
be the probability that you

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lose times the utility if you
lose plus the probability that

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you win times the utility
if you win.

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And utility is not the same as
value, importantly, because

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utility functions exhibit
diminishing marginal utility.

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Utility functions
are not linear.

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Utility functions
are nonlinear.

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And, in particular, there's
diminishing marginal utility.

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And with diminishing marginal
utility, you're going to not

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want bets where there's the
chance you lose is equal to or

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even a bit smaller than the
value that you win.

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And the basic point is that the
joy of winning is smaller

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than the pain of losing with
diminishing marginal utility.

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Yeah.

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AUDIENCE: Isn't there also a
statistical side to this then?

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Because we don't know how many
times we're going to bet.

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It might just be once.

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We're a lot more comfortable if,
let's say, use the law of

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large numbers and say, OK, it's
going to eventually even

00:06:29.332 --> 00:06:30.992
out so we'll win
$12.50 a game.

00:06:30.992 --> 00:06:34.277
But for the first, let's say 10
or so games, we might get

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really unlucky and flip eight
tails and two heads.

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PROFESSOR: But, once again, if
you weren't risk averse, you

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wouldn't care about that.

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Hold that thought.

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I'm going to explain why
that isn't true.

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So just hold that thought.

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So now let's imagine that your
utility functions are the

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typical form we've worked
with before, the typical

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diminishing marginal utility
form we've worked with before

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where utility is the square
root of consumption.

00:06:56.130 --> 00:06:58.940
You're casting your mind back
to consumer theory here.

00:06:58.940 --> 00:07:00.640
You're going to have to start
integrating the course now,

00:07:00.640 --> 00:07:02.300
both consumer and
producer theory.

00:07:02.300 --> 00:07:04.700
So remember we said the typical
diminishing marginal

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utility function we worked
with was u equals the

00:07:08.430 --> 00:07:09.680
square root of c.

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Now, let's say you start with
consumption of $100.

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Imagine you consume
your income.

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Let's say you have consumption
of $100.

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Well, then utility is 10.

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If you start with a consumption
of $100, your

00:07:24.130 --> 00:07:26.530
utility is 10.

00:07:26.530 --> 00:07:28.090
Now let's calculate
the expected

00:07:28.090 --> 00:07:30.560
utility of this gamble.

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The expected utility of this
gamble is that there's a 50%

00:07:34.090 --> 00:07:37.240
chance that you lose.

00:07:37.240 --> 00:07:39.250
And, if you lose, what
is your utility?

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Well you lose $100.

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So consumption goes to 0.

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So utility is 0 plus a 50%
chance that you win.

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Well, what do you
get if you win.

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Well, if you win, you go
from $100 to $225.

00:07:54.510 --> 00:07:59.600
So your utility is the square
root of $225, or $15.

00:07:59.600 --> 00:08:04.720
Your utility is the square
root of $225 or 15.

00:08:04.720 --> 00:08:08.590
It's half chance of having 15.

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I'm sorry.

00:08:09.000 --> 00:08:10.780
So this is a negative.

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Yeah.

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So utility, if you take this
gamble, is you end up with a

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utility of 7.5.

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So utility falls.

00:08:18.000 --> 00:08:20.960
You move from a utility of 10
without the gamble to a

00:08:20.960 --> 00:08:23.730
utility of 7.5 with
the gamble.

00:08:23.730 --> 00:08:26.520
Utility is lower with the
gamble, which is why people

00:08:26.520 --> 00:08:29.080
decided they didn't want
to take that gamble.

00:08:29.080 --> 00:08:31.630
Utility is lower.

00:08:31.630 --> 00:08:34.789
And the reason is because given
a utility function of

00:08:34.789 --> 00:08:38.840
this form, you are sadder about
losing than happier

00:08:38.840 --> 00:08:41.530
about winning.

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To see that, we can see that
graphically in Figure 20-1.

00:08:46.680 --> 00:08:50.510
This graph's utility against
wealth-- we don't usually

00:08:50.510 --> 00:08:52.370
graph utility, because
it's not cardinal.

00:08:52.370 --> 00:08:53.110
Remember it's just ordinal.

00:08:53.110 --> 00:08:55.050
But the sort of gives you a
sense of the intuition.

00:08:55.050 --> 00:08:59.020
This is a graph of utility
against wealth levels.

00:08:59.020 --> 00:09:06.710
So you start at point A. You
start with $100 in wealth,

00:09:06.710 --> 00:09:08.000
which is consumption.

00:09:08.000 --> 00:09:10.390
and utility of 10.

00:09:10.390 --> 00:09:14.110
Now, I give you a choice
of a gamble.

00:09:14.110 --> 00:09:18.700
That gamble has a 50% chance of
leaving you at 0 and a 50%

00:09:18.700 --> 00:09:23.570
chance a leaving you at point
B. So your utility and

00:09:23.570 --> 00:09:26.950
expected value is the midpoint
of that chord that runs from 0

00:09:26.950 --> 00:09:32.530
to B or point C. Your expecting
utility is lower

00:09:32.530 --> 00:09:35.370
than your initial utility.

00:09:35.370 --> 00:09:35.830
Why?

00:09:35.830 --> 00:09:40.190
Because utility is concave. You
are made so sad by getting

00:09:40.190 --> 00:09:44.250
to 0 that it vastly
overcompensate the happiness

00:09:44.250 --> 00:09:47.900
you feel moving to $225 because
of the diminishing

00:09:47.900 --> 00:09:49.910
marginal utility.

00:09:49.910 --> 00:09:51.610
Because, basically, think
of it this way.

00:09:51.610 --> 00:09:53.290
Imagine it's your
actual income.

00:09:53.290 --> 00:09:55.450
Let's take the point about the
size of the gamble relative to

00:09:55.450 --> 00:09:56.340
income seriously.

00:09:56.340 --> 00:09:58.250
Imagine, literally, I was asking
you to gamble your

00:09:58.250 --> 00:09:59.820
entire income for the year.

00:09:59.820 --> 00:10:02.150
And if you lose, you
starve to death.

00:10:02.150 --> 00:10:04.070
And if you win, you get
to eat extra nice.

00:10:04.070 --> 00:10:08.360
Well, clearly, the disutility
of starving to death vastly

00:10:08.360 --> 00:10:11.590
outweighs the extra utility
to eating well.

00:10:11.590 --> 00:10:13.950
So, in that extreme example,
if this was your entire

00:10:13.950 --> 00:10:17.190
wealth, you can see why you
would have a situation where

00:10:17.190 --> 00:10:18.820
you wouldn't want to
take that gamble.

00:10:18.820 --> 00:10:21.760
Because if you lost,
you'd die.

00:10:21.760 --> 00:10:27.110
And, basically, risk aversion
arises because, basically,

00:10:27.110 --> 00:10:29.410
with diminishing marginal
utility you're

00:10:29.410 --> 00:10:30.730
made so much sadder.

00:10:30.730 --> 00:10:34.260
That steepness at the bottom,
you get so much sadder as you

00:10:34.260 --> 00:10:39.290
get towards 0 that it vastly
overcompensates the flatter

00:10:39.290 --> 00:10:43.030
part as you move above
your initial point.

00:10:43.030 --> 00:10:46.150
So, as you can see, you are
going to end up not wanting

00:10:46.150 --> 00:10:48.110
gambles even if they're fair.

00:10:48.110 --> 00:10:53.230
Gambles that are fair, that is
positive expected value, might

00:10:53.230 --> 00:10:57.860
still lead to a reduction in
your expected utility.

00:10:57.860 --> 00:10:59.760
Indeed, let me go further.

00:10:59.760 --> 00:11:03.310
You dislike this gamble so
much that if I said the

00:11:03.310 --> 00:11:06.840
following, I as your teacher am
going to force you to take

00:11:06.840 --> 00:11:09.740
this gamble-- imagine it's
like 100 years ago where

00:11:09.740 --> 00:11:11.830
teachers can beat students
and stuff--

00:11:11.830 --> 00:11:17.170
I'm going to force you take this
gamble unless you pay me,

00:11:17.170 --> 00:11:20.240
you would actually be
willing to pay me to

00:11:20.240 --> 00:11:23.310
avoid taking this gamble.

00:11:23.310 --> 00:11:24.560
How much would you pay me?

00:11:28.110 --> 00:11:30.890
Imagine utilities in dollar
terms. Imagine we're actually

00:11:30.890 --> 00:11:33.720
measuring utility in dollar
terms. How much would you pay

00:11:33.720 --> 00:11:37.160
me to avoid taking
this gamble.

00:11:37.160 --> 00:11:41.790
If I said you either take the
gamble, or you pay me.

00:11:41.790 --> 00:11:44.300
You're starting with
a utility of 100.

00:11:44.300 --> 00:11:44.800
Yeah?

00:11:44.800 --> 00:11:47.800
AUDIENCE: The difference between
the two utilities.

00:11:47.800 --> 00:11:48.550
PROFESSOR: Well, the difference

00:11:48.550 --> 00:11:49.530
between the two utilities.

00:11:49.530 --> 00:11:52.250
So utility is 100 here.

00:11:52.250 --> 00:11:56.740
Here utility is 7.5
squared, so 56.25.

00:11:56.740 --> 00:12:01.020
So you would actually
pay me $43.75 to

00:12:01.020 --> 00:12:02.090
avoid taking this gamble.

00:12:02.090 --> 00:12:03.120
Think about that.

00:12:03.120 --> 00:12:08.230
I've offered you a more than
fair bet, a very good bet,

00:12:08.230 --> 00:12:11.220
which, on average, will yield
you a positive $12.50.

00:12:11.220 --> 00:12:14.720
Yet you will pay me $43.75.

00:12:14.720 --> 00:12:18.390
You will pay almost half of your
entire wealth to avoid

00:12:18.390 --> 00:12:20.210
taking that gamble.

00:12:20.210 --> 00:12:22.300
That's pretty incredible
if you think about it.

00:12:22.300 --> 00:12:25.600
I've offered you a more than
fair bet, and yet you will pay

00:12:25.600 --> 00:12:28.390
me more than half your wealth,
almost half your wealth, to

00:12:28.390 --> 00:12:31.850
avoid taking that bet.

00:12:31.850 --> 00:12:32.960
So another way to see
this, let's look at

00:12:32.960 --> 00:12:33.930
this another way.

00:12:33.930 --> 00:12:37.800
How large would I have to make
the positive payoff for you to

00:12:37.800 --> 00:12:39.085
take the bet?

00:12:39.085 --> 00:12:41.120
Let's look at it that way.

00:12:41.120 --> 00:12:43.700
Right now I said you win
$125 with heads.

00:12:43.700 --> 00:12:46.853
How much would you have to win
with heads if you were going

00:12:46.853 --> 00:12:47.320
to take that bet?

00:12:47.320 --> 00:12:48.480
Yeah.

00:12:48.480 --> 00:12:50.267
And tell us how you
figured that out.

00:12:50.267 --> 00:12:53.000
AUDIENCE: Because you need to
have at least the same utility

00:12:53.000 --> 00:12:55.734
as you had before from the
unexpected utility.

00:12:55.734 --> 00:12:59.710
So more than half of his
per year utility

00:12:59.710 --> 00:13:01.718
would be 20 if he wins.

00:13:01.718 --> 00:13:03.510
20 squared is 400.

00:13:03.510 --> 00:13:04.860
[INAUDIBLE  PHRASE].

00:13:04.860 --> 00:13:05.090
PROFESSOR: Right.

00:13:05.090 --> 00:13:06.570
You'd need to win 300.

00:13:06.570 --> 00:13:10.170
Because I'd need to take your
utility to 20 if you win.

00:13:10.170 --> 00:13:13.150
Only then would you be willing
to take this gamble.

00:13:13.150 --> 00:13:16.760
So another way to say it is
that's how fair a gamble would

00:13:16.760 --> 00:13:18.436
need to be, how more than
fair it would need to be

00:13:18.436 --> 00:13:19.380
before you take it.

00:13:19.380 --> 00:13:25.100
You'd need me to pay off 3:1 on
a 50% chance before you'd

00:13:25.100 --> 00:13:27.080
take the bet.

00:13:27.080 --> 00:13:29.020
And this is just with a typical
looking utility

00:13:29.020 --> 00:13:31.510
function of the kind we worked
earlier in the semester.

00:13:31.510 --> 00:13:32.290
You didn't look at this earlier
in the semester and

00:13:32.290 --> 00:13:34.510
say, wow, that's a bizarre
utility function.

00:13:34.510 --> 00:13:37.080
We got sensible answers on our
problems, and problem sets,

00:13:37.080 --> 00:13:38.950
and tests, and things,
examples from

00:13:38.950 --> 00:13:39.520
square root of c.

00:13:39.520 --> 00:13:41.490
That seemed like a sensible
function.

00:13:41.490 --> 00:13:45.580
And yet it yields these
incredibly wild predictions

00:13:45.580 --> 00:13:48.220
that you would pay people almost
half of your wealth to

00:13:48.220 --> 00:13:50.330
avoid engaging in a more
than fair bet.

00:13:53.890 --> 00:13:57.920
And that you would need the odds
to be like 3:1 before you

00:13:57.920 --> 00:13:59.820
even consider taking a a bet.

00:13:59.820 --> 00:14:02.150
That's the power of uncertainty
and the power of

00:14:02.150 --> 00:14:03.140
risk aversion.

00:14:03.140 --> 00:14:06.020
Really, risk aversion, it's just
the power of diminishing

00:14:06.020 --> 00:14:08.410
marginal utility.

00:14:08.410 --> 00:14:11.120
The power of diminishing
marginal utility is so key to

00:14:11.120 --> 00:14:12.760
driving our decisions.

00:14:12.760 --> 00:14:15.870
It's the fact that that first
pizza means so much more to

00:14:15.870 --> 00:14:20.530
you than the fifth pizza, that
you really hate outcomes that

00:14:20.530 --> 00:14:22.680
don't let you get
the first pizza.

00:14:22.680 --> 00:14:27.050
And, as a result, you will pay
a lot to be forced into a

00:14:27.050 --> 00:14:29.970
situation where you don't
get any pizzas.

00:14:29.970 --> 00:14:32.580
You'll need to be paid a lot in
the state where you do win

00:14:32.580 --> 00:14:35.670
to deal with the state
where you don't.

00:14:35.670 --> 00:14:36.920
Questions about that?

00:14:39.310 --> 00:14:41.570
Now, we can change the example
in some interesting ways to

00:14:41.570 --> 00:14:42.200
understand it.

00:14:42.200 --> 00:14:46.920
So let's change the example to
say, instead, let's talk about

00:14:46.920 --> 00:14:51.510
some alternatives to this
example and how they affect

00:14:51.510 --> 00:14:52.330
our intuition.

00:14:52.330 --> 00:14:55.940
First alternative, imagine your
utility function instead

00:14:55.940 --> 00:15:00.040
of being square root of c, your
utility function was 0.1

00:15:00.040 --> 00:15:05.270
times c, a linear utility
function, not a non-linear

00:15:05.270 --> 00:15:06.520
utility function.

00:15:11.670 --> 00:15:16.360
We can now say that, in that
case, you actually would take

00:15:16.360 --> 00:15:17.610
the gamble.

00:15:20.780 --> 00:15:23.190
There's a 0.5% chance of 0.

00:15:26.340 --> 00:15:28.780
And I chose 0.1 times c, because
your initial utility

00:15:28.780 --> 00:15:31.020
is still 10 then.

00:15:31.020 --> 00:15:31.890
I normalized this.

00:15:31.890 --> 00:15:36.950
So starting with your bundle of
100 you still start at 10.

00:15:36.950 --> 00:15:38.080
It gives the same starting
point as

00:15:38.080 --> 00:15:40.560
the square root function.

00:15:40.560 --> 00:15:45.200
But now your expected utility
from his gamble is 0.5 times 0

00:15:45.200 --> 00:15:53.950
plus 0.5 times if you win 125,
your utility is 12.5.

00:15:53.950 --> 00:15:54.460
I'm sorry.

00:15:54.460 --> 00:15:59.020
It's 22.5.

00:15:59.020 --> 00:16:05.410
So your expected utility is
11.25 which is higher than

00:16:05.410 --> 00:16:07.530
your starting utility.

00:16:07.530 --> 00:16:10.210
So you would take this gamble.

00:16:10.210 --> 00:16:12.280
What's changed?

00:16:12.280 --> 00:16:13.660
AUDIENCE: No diminishing
marginal utility.

00:16:13.660 --> 00:16:15.880
PROFESSOR: No diminishing
marginal utility because now

00:16:15.880 --> 00:16:17.310
we are no longer risk averse.

00:16:17.310 --> 00:16:21.050
We are what we call
risk neutral.

00:16:21.050 --> 00:16:26.210
A linear utility function
yields risks neutrality.

00:16:26.210 --> 00:16:29.910
And once you're risk neutral,
you only care

00:16:29.910 --> 00:16:33.790
about expected value.

00:16:33.790 --> 00:16:36.700
Risk neutral consumers
would only care

00:16:36.700 --> 00:16:39.320
about expected value.

00:16:39.320 --> 00:16:41.650
And so a linear utility function
will lead to risk

00:16:41.650 --> 00:16:42.410
neutrality since you don't have

00:16:42.410 --> 00:16:43.450
diminishing marginal utility.

00:16:43.450 --> 00:16:44.605
Then you take any
bet that's fair.

00:16:44.605 --> 00:16:45.460
You don't care.

00:16:45.460 --> 00:16:47.820
You're indifferent between
winning a dollar and losing a

00:16:47.820 --> 00:16:48.960
dollar with this utility
function.

00:16:48.960 --> 00:16:53.200
It doesn't matter if
you go up or down.

00:16:53.200 --> 00:16:55.100
The joy you get from winning
is the same as the pain you

00:16:55.100 --> 00:16:56.170
get from losing.

00:16:56.170 --> 00:16:58.090
Whereas with this utility
function, the pain you get

00:16:58.090 --> 00:17:01.610
from losing exceeds the
joy from winning.

00:17:01.610 --> 00:17:07.180
We can see that graphically in
the next figure, Figure 20-2,

00:17:07.180 --> 00:17:09.500
the case of risk neutrality.

00:17:09.500 --> 00:17:16.359
Here, you start at point
A. You have 100, and

00:17:16.359 --> 00:17:19.900
your utility is 10.

00:17:19.900 --> 00:17:22.220
Now, I've offered you a gamble
where there's a 50% chance of

00:17:22.220 --> 00:17:27.170
getting 0 and a 50% chance of
getting B. Well, that yields

00:17:27.170 --> 00:17:32.030
an outcome of c, which
is a higher utility.

00:17:32.030 --> 00:17:36.290
So since your utility is linear,
you're risk neutral,

00:17:36.290 --> 00:17:37.830
and you'll take any fair bet.

00:17:40.830 --> 00:17:42.440
We can go further.

00:17:42.440 --> 00:17:50.090
What if utility, instead, was of
the form u equals c squared

00:17:50.090 --> 00:17:52.760
over 1,000?

00:17:52.760 --> 00:17:54.810
What if this was your
utility function?

00:17:54.810 --> 00:17:59.280
Once again, your initial
utility u of 100 is 10.

00:17:59.280 --> 00:18:02.850
It's the same starting point.

00:18:02.850 --> 00:18:07.440
But this is a utility function
which now if you do this

00:18:07.440 --> 00:18:17.560
gamble, your expected utility is
50% times 0 plus 50% times

00:18:17.560 --> 00:18:30.100
$225 squared over 1,000
which is 25.3.

00:18:30.100 --> 00:18:34.460
That's a huge increase in
utility from this gamble.

00:18:34.460 --> 00:18:39.880
So your expected utility with
the gamble is 25.3.

00:18:39.880 --> 00:18:43.480
It's a huge increase
in utility.

00:18:43.480 --> 00:18:47.830
And that's because this is an
individual where the shape of

00:18:47.830 --> 00:18:49.860
the utility function has change
where they don't have

00:18:49.860 --> 00:18:51.320
diminishing marginal
utility, they have

00:18:51.320 --> 00:18:53.770
increasing marginal utility.

00:18:53.770 --> 00:18:54.650
We've never worked with utility

00:18:54.650 --> 00:18:56.380
functions like this before.

00:18:56.380 --> 00:19:00.280
These are individuals
we call risk-loving.

00:19:00.280 --> 00:19:04.810
That is, they are made happier
by winning $1 than they are

00:19:04.810 --> 00:19:06.760
made sadder by losing $1.

00:19:06.760 --> 00:19:09.030
It's the opposite of all the
intuition we developed earlier

00:19:09.030 --> 00:19:09.970
in this course.

00:19:09.970 --> 00:19:12.550
It's a crazy utility function.

00:19:12.550 --> 00:19:14.960
But the notion of a risk-loving
utility function

00:19:14.960 --> 00:19:18.375
is one where literally $1 that
moves you up makes you happier

00:19:18.375 --> 00:19:20.630
than $1 that moves you down
makes you sadder.

00:19:20.630 --> 00:19:27.650
You can see that
in Figure 20-3.

00:19:27.650 --> 00:19:30.550
Here's a risk-loving
utility function.

00:19:30.550 --> 00:19:37.770
The individual starts at point
A. They have a choice of a

00:19:37.770 --> 00:19:44.400
gamble where they can have a 50%
chance of landing at 0 and

00:19:44.400 --> 00:19:45.330
a 50% chance--

00:19:45.330 --> 00:19:46.800
Jessica, that B should
be down at the

00:19:46.800 --> 00:19:48.550
intersection of dashed lines--

00:19:48.550 --> 00:19:51.305
a 50% chance of landing at B
at the intersection of the

00:19:51.305 --> 00:19:52.770
dashed lines.

00:19:52.770 --> 00:19:55.220
You take the average of
those two, and it's c.

00:19:55.220 --> 00:19:58.780
Their utility is way higher with
the gamble than it was

00:19:58.780 --> 00:20:00.870
without the gamble.

00:20:00.870 --> 00:20:01.560
In fact.

00:20:01.560 --> 00:20:03.540
we can go further.

00:20:03.540 --> 00:20:06.390
With a risk-loving person, they
would actually take an

00:20:06.390 --> 00:20:08.650
unfair bet.

00:20:08.650 --> 00:20:10.970
Consider the following bet.

00:20:10.970 --> 00:20:16.230
Tails you lose $100,
heads you win $75.

00:20:16.230 --> 00:20:18.455
That's a bet with a negative
expected value.

00:20:22.900 --> 00:20:25.520
Neither the risk averse nor the
risk neutral person would

00:20:25.520 --> 00:20:26.320
take that bet.

00:20:26.320 --> 00:20:27.490
But a risk-loving
person would.

00:20:27.490 --> 00:20:31.610
If you work out the math, that
bet gives them a gain in

00:20:31.610 --> 00:20:33.080
expected utility.

00:20:33.080 --> 00:20:35.380
That is a bet with a negative
expected value that gives them

00:20:35.380 --> 00:20:36.740
a gain in expected utility.

00:20:36.740 --> 00:20:37.845
Why is that?

00:20:37.845 --> 00:20:39.610
Because it's the opposite
of diminishing

00:20:39.610 --> 00:20:40.960
marginal utility intuition.

00:20:40.960 --> 00:20:45.740
They're made so much happier
by winning that they're

00:20:45.740 --> 00:20:46.910
willing to take a bet
even if it's a

00:20:46.910 --> 00:20:47.990
negative expected value.

00:20:47.990 --> 00:20:50.220
Just like the risk averse person
is made so much sadder

00:20:50.220 --> 00:20:52.310
by losing, they won't
take a bet even if

00:20:52.310 --> 00:20:54.000
it's more than fair.

00:20:54.000 --> 00:20:57.650
So you can actually develop all
the opposite predictions

00:20:57.650 --> 00:20:58.880
from a risk-loving person.

00:20:58.880 --> 00:21:02.150
They'll even take
unfair gambles.

00:21:02.150 --> 00:21:03.830
Now, by the way, I skipped over
your earlier question

00:21:03.830 --> 00:21:04.910
about risk neutrality.

00:21:04.910 --> 00:21:07.700
With risk neutrality, you see it
doesn't matter if you do it

00:21:07.700 --> 00:21:09.920
100 times or one time.

00:21:09.920 --> 00:21:13.860
If you're risk neutral, you
should take the bet anytime,

00:21:13.860 --> 00:21:16.530
because the expected value
is still positive.

00:21:16.530 --> 00:21:18.145
Now, you're thinking about
risk aversion where, in

00:21:18.145 --> 00:21:21.100
substance, you're more
confident as

00:21:21.100 --> 00:21:22.150
the numbers go up.

00:21:22.150 --> 00:21:24.380
But if you're risk neutral,
you'll take it no matter how

00:21:24.380 --> 00:21:27.090
many times I offer
you that bet.

00:21:27.090 --> 00:21:32.870
So to extend this further, let's
go to a third extension

00:21:32.870 --> 00:21:35.460
which will develop this
intuition further.

00:21:35.460 --> 00:21:39.000
Now imagine that I offer you
guys a different gamble.

00:21:39.000 --> 00:21:40.960
And, once again, I really want
you to answer honestly.

00:21:40.960 --> 00:21:41.925
Don't try to game me.

00:21:41.925 --> 00:21:43.120
Answer honestly.

00:21:43.120 --> 00:21:54.380
Now the gamble is if I flip a
coin, tails you lose $1, heads

00:21:54.380 --> 00:21:57.510
you win $1.25.

00:21:57.510 --> 00:22:00.470
Now how many of you would
take that gamble.

00:22:00.470 --> 00:22:02.770
How many would not
take that gamble?

00:22:02.770 --> 00:22:03.750
OK.

00:22:03.750 --> 00:22:04.940
I hope you're answering
honestly.

00:22:04.940 --> 00:22:07.870
But maybe you're just thinking
ahead and realizing that that

00:22:07.870 --> 00:22:08.680
gamble is very different.

00:22:08.680 --> 00:22:11.350
And why are people more willing
to take that gamble

00:22:11.350 --> 00:22:13.710
than they were willing to take
the previous gamble, the same

00:22:13.710 --> 00:22:14.390
risk averse people.

00:22:14.390 --> 00:22:15.156
Yeah.

00:22:15.156 --> 00:22:19.440
AUDIENCE: The difference
in [INAUDIBLE PHRASE].

00:22:19.440 --> 00:22:19.970
PROFESSOR: Exactly.

00:22:19.970 --> 00:22:23.535
In particular, the utility
function is locally linear.

00:22:28.350 --> 00:22:31.400
Let's go back to Figure 20-1.

00:22:31.400 --> 00:22:34.760
As you get closer and closer
to A, you could draw,

00:22:34.760 --> 00:22:37.300
essentially, a linear segment.

00:22:37.300 --> 00:22:42.810
So for an infinitesimal bet,
utility is linear.

00:22:42.810 --> 00:22:45.080
So it's linear at point A.

00:22:45.080 --> 00:22:48.740
So for small bets, you
become risk neutral.

00:22:48.740 --> 00:22:52.600
Even a risk averse person moves
towards risk neutrality

00:22:52.600 --> 00:22:55.750
as the bet is small relative
to their resources.

00:22:55.750 --> 00:22:57.380
This was the point that
you were making.

00:22:57.380 --> 00:22:59.140
Basically if you're a rich
person, you'd probably be

00:22:59.140 --> 00:23:01.280
happy to take the $100
and $125 thing.

00:23:01.280 --> 00:23:02.280
I'd be happy to do that.

00:23:02.280 --> 00:23:04.310
I'm a rich guy.

00:23:04.310 --> 00:23:05.890
I'd be happy to do that.

00:23:05.890 --> 00:23:10.550
So, basically, what determines
your willingness to take a bet

00:23:10.550 --> 00:23:12.880
is going to be about
what's at stake

00:23:12.880 --> 00:23:14.130
relative to your resources.

00:23:17.660 --> 00:23:21.500
And what you can see is that
if you solve the math here,

00:23:21.500 --> 00:23:24.130
that basically expected utility
even with a square

00:23:24.130 --> 00:23:30.890
root of c is positive for
that smaller gamble.

00:23:30.890 --> 00:23:33.830
Because as it gets smaller
relative to the $100 you start

00:23:33.830 --> 00:23:36.460
with, you become roughly
risk neutral.

00:23:36.460 --> 00:23:39.250
And then you'll go ahead
and take the gamble.

00:23:39.250 --> 00:23:42.750
So at the end of the day what's
going to determine

00:23:42.750 --> 00:23:47.030
whether you're going to take a
gamble is going to be your

00:23:47.030 --> 00:23:53.090
level of risk aversion and the
size of the risk you're taking

00:23:53.090 --> 00:23:55.140
relative to your resources.

00:23:55.140 --> 00:23:59.110
The more risk averse you are,
and the bigger the gamble, the

00:23:59.110 --> 00:24:01.315
less likely you are to take it
at a given level of fairness.

00:24:04.530 --> 00:24:05.780
Questions about that?

00:24:08.490 --> 00:24:09.210
All right.

00:24:09.210 --> 00:24:12.640
So now that we all understand
expected utility theory.

00:24:12.640 --> 00:24:14.670
Now we're going to go on and
talk about why this matters in

00:24:14.670 --> 00:24:16.340
the real world and
how we use it.

00:24:16.340 --> 00:24:17.940
And I want to talk, in
particular, about two

00:24:17.940 --> 00:24:22.880
applications, insurance
and the lottery.

00:24:22.880 --> 00:24:25.630
Let's start by talking
about insurance and

00:24:25.630 --> 00:24:27.640
why people have insurance.

00:24:27.640 --> 00:24:31.150
Because, in fact, given what
we learned in this lecture,

00:24:31.150 --> 00:24:33.250
there would be no reason
for insurance.

00:24:33.250 --> 00:24:35.336
This lecture tells us why
people have insurance.

00:24:37.910 --> 00:24:40.510
Because there's diminishing
marginal utility, and you're

00:24:40.510 --> 00:24:43.100
made so much sadder with a
negative outcome, you're

00:24:43.100 --> 00:24:45.560
willing to pay you avoid it.

00:24:45.560 --> 00:24:47.280
Remember we talked about that
you would be willing to pay

00:24:47.280 --> 00:24:50.400
almost $44 to avoid being
forced to take that bet?

00:24:50.400 --> 00:24:52.580
That's what insurance does.

00:24:52.580 --> 00:24:56.230
Insurance allows you to
avoid taking gambles.

00:24:56.230 --> 00:24:57.390
That's what you can think
of insurance as.

00:24:57.390 --> 00:24:59.300
It's a way to avoid
taking a gamble.

00:24:59.300 --> 00:25:01.130
You're gambling you're
going to get sick.

00:25:01.130 --> 00:25:03.670
You're gambling your house
is going to burn down.

00:25:03.670 --> 00:25:05.060
These are gambles you
face that are

00:25:05.060 --> 00:25:07.590
forced on you by nature.

00:25:07.590 --> 00:25:09.500
What insurance does is
allow you to avoid

00:25:09.500 --> 00:25:10.530
taking those gambles.

00:25:10.530 --> 00:25:13.960
And just like you'd pay me to
avoid the $100, $125 gamble,

00:25:13.960 --> 00:25:16.940
you're paying Aetna to avoid
gambling that you might have

00:25:16.940 --> 00:25:19.190
to go to the hospital.

00:25:19.190 --> 00:25:21.990
So let's say there's a
25-year-old who is deciding

00:25:21.990 --> 00:25:23.470
whether to buy health
insurance.

00:25:23.470 --> 00:25:24.760
And let's say they're
25-year-old

00:25:24.760 --> 00:25:26.320
guy, totally healthy.

00:25:26.320 --> 00:25:27.490
I say guy because there's
no risk they're

00:25:27.490 --> 00:25:28.570
going to have a kid.

00:25:28.570 --> 00:25:32.130
So he's basically totally
healthy, basically zero chance

00:25:32.130 --> 00:25:33.422
they're going to use the
doctor except if they

00:25:33.422 --> 00:25:35.470
get hit by a car.

00:25:35.470 --> 00:25:38.420
So imagine the situation is that
you've got 25-year-old

00:25:38.420 --> 00:25:45.440
with an income of $40,000.

00:25:45.440 --> 00:25:48.320
And let's say that there's
a 1% chance that they'll

00:25:48.320 --> 00:25:48.960
get hit by a car.

00:25:48.960 --> 00:25:50.340
It is Cambridge after all.

00:25:50.340 --> 00:25:52.190
So every time you cross the
street, there's a 1% chance

00:25:52.190 --> 00:25:53.440
you get hit by a car.

00:25:57.380 --> 00:26:00.440
And if you get hit by a car,
you're going to suffer $30,000

00:26:00.440 --> 00:26:01.690
in hospital bills.

00:26:09.480 --> 00:26:13.060
And let's say your utility
function is square root of c.

00:26:13.060 --> 00:26:14.310
So you're a risk averse guy.

00:26:17.270 --> 00:26:24.260
So let's say that I then come
to you and say, look, each

00:26:24.260 --> 00:26:30.320
year there's an expected
cost to you of getting

00:26:30.320 --> 00:26:32.250
hit by car of $300.

00:26:32.250 --> 00:26:33.160
How did I calculate that?

00:26:33.160 --> 00:26:36.280
Well, every year there's a
1% chance you get hit.

00:26:36.280 --> 00:26:38.020
They're independent
draws, let's say.

00:26:38.020 --> 00:26:39.600
If you get hit this year,
it doesn't mean

00:26:39.600 --> 00:26:41.000
suddenly you're safer.

00:26:41.000 --> 00:26:41.710
It's random.

00:26:41.710 --> 00:26:43.140
It's just crazy drivers.

00:26:43.140 --> 00:26:45.630
So there's a 1% chance you're
going to get hit every year.

00:26:45.630 --> 00:26:48.133
And if you get hit, there's a
$30,000 cost. So every year

00:26:48.133 --> 00:26:49.880
there's an expected
cost to you--

00:26:49.880 --> 00:26:51.870
the opposite of expected value
is expected cost--

00:26:51.870 --> 00:26:53.590
of $300.

00:26:53.590 --> 00:26:55.910
So let's say I offered to sell
you insurance for $300.

00:26:55.910 --> 00:26:59.740
I offered to sell you insurance
in a way where, on

00:26:59.740 --> 00:27:04.890
average, if you lived an
infinite number of years, you

00:27:04.890 --> 00:27:08.020
would pay out in premiums what
you'd get in benefits.

00:27:08.020 --> 00:27:11.080
If you paid $300 a year and
lived forever or lived for

00:27:11.080 --> 00:27:14.520
many, many years-- the law of
large numbers enough years--

00:27:14.520 --> 00:27:16.460
then basically you would pay out
in premiums what you would

00:27:16.460 --> 00:27:18.850
collect in benefits.

00:27:18.850 --> 00:27:20.830
You'd get hit once
every 100 years.

00:27:20.830 --> 00:27:23.850
And ever 100 years you would
have paid $30,000 in premiums,

00:27:23.850 --> 00:27:26.330
and you'd collect $30,000
in benefits.

00:27:26.330 --> 00:27:30.880
So that's what we call
actuarially fair insurance.

00:27:30.880 --> 00:27:43.030
Actuarially fair insurance is
insurance where the price of

00:27:43.030 --> 00:27:50.560
the insurance equals the
probability of the bad outcome

00:27:50.560 --> 00:27:54.770
times the cost of
the bad outcome.

00:27:57.750 --> 00:28:00.140
That's actuarially fair
insurance where the price you

00:28:00.140 --> 00:28:02.005
pay is the probability of the
bad outcome times the cost of

00:28:02.005 --> 00:28:02.550
the bad outcome.

00:28:02.550 --> 00:28:05.540
That's fair because, over a
large enough population, the

00:28:05.540 --> 00:28:10.250
premiums that get paid in will
get paid out in the form of

00:28:10.250 --> 00:28:13.120
claims.

00:28:13.120 --> 00:28:18.460
Now, let's ask what is your
utility if you do not or do

00:28:18.460 --> 00:28:19.190
buy insurance.

00:28:19.190 --> 00:28:20.780
So for the first thing you
say if I'm a 25-year-old.

00:28:20.780 --> 00:28:21.240
Screw it.

00:28:21.240 --> 00:28:22.170
I'm never going to
get hit by a car.

00:28:22.170 --> 00:28:23.430
I'm not going to
buy insurance.

00:28:23.430 --> 00:28:26.680
What's your utility
with no insurance?

00:28:26.680 --> 00:28:32.410
Well, if you have no insurance,
there's a 1%

00:28:32.410 --> 00:28:36.020
chance, 0.01, that you'll
lose $30,000.

00:28:36.020 --> 00:28:37.305
You'll get hit by a car
and lose $30,000.

00:28:37.305 --> 00:28:39.710
Your income is $40,000.

00:28:39.710 --> 00:28:42.890
So there's a 1% chance that
you'll end up with a utility,

00:28:42.890 --> 00:28:47.910
which is the square
root of 10,000.

00:28:47.910 --> 00:28:55.850
And there's a 99% chance you'll
end up with a utility

00:28:55.850 --> 00:29:00.120
that's the square
root of 40,000.

00:29:00.120 --> 00:29:09.200
You work this out, and the
answer is you get 199.

00:29:09.200 --> 00:29:14.700
Utility without insurance is 199
which is pretty close to

00:29:14.700 --> 00:29:16.510
utility just if you weren't
going to get hit by the car.

00:29:16.510 --> 00:29:18.660
Because it's so rare that
you get hit by the car.

00:29:18.660 --> 00:29:23.160
So utility is 199 without
insurance.

00:29:23.160 --> 00:29:30.170
Now, let's ask the question, how
much would you be willing

00:29:30.170 --> 00:29:33.520
to pay to have insurance?

00:29:33.520 --> 00:29:34.980
How do we figure that out?

00:29:34.980 --> 00:29:37.560
$300 is the actuarially
fair premium.

00:29:37.560 --> 00:29:38.500
But now let's do a different
question.

00:29:38.500 --> 00:29:40.560
I'm an insurance company, and
I want to make money.

00:29:40.560 --> 00:29:42.160
I don't want to just charge the
actuarially fair premium.

00:29:42.160 --> 00:29:43.770
The insurance company
makes no money with

00:29:43.770 --> 00:29:45.400
this premium of $300.

00:29:45.400 --> 00:29:47.870
So the insurance company
wants to make money.

00:29:47.870 --> 00:29:50.010
How would we figure out how much
would you be willing to

00:29:50.010 --> 00:29:52.340
pay, this 25-year-old,
be willing

00:29:52.340 --> 00:29:54.800
to pay to get insurance?

00:29:54.800 --> 00:29:57.590
How do we figure that out?

00:29:57.590 --> 00:29:58.950
Yeah.

00:29:58.950 --> 00:30:02.020
AUDIENCE: Maybe you could keep
the utility function constant.

00:30:02.020 --> 00:30:03.315
PROFESSOR: Keep the utility
value constant.

00:30:03.315 --> 00:30:04.140
AUDIENCE: Value, yes.

00:30:04.140 --> 00:30:04.730
PROFESSOR: Exactly.

00:30:04.730 --> 00:30:08.520
You'd have ask well, how much
would I be willing to pay to

00:30:08.520 --> 00:30:10.420
have insurance which would
protect me and leave me at the

00:30:10.420 --> 00:30:12.190
same utility level.

00:30:12.190 --> 00:30:13.320
Obviously it would have to
be a little bit higher.

00:30:13.320 --> 00:30:15.770
But let's just set it equal.

00:30:15.770 --> 00:30:19.770
So, in other words, if I bought
insurance, my utility

00:30:19.770 --> 00:30:28.200
with insurance, there's a
1% chance that I will

00:30:28.200 --> 00:30:29.490
get hit by the car.

00:30:29.490 --> 00:30:30.980
In that case, what
happens to me?

00:30:30.980 --> 00:30:36.290
Well, if I get hit by the
car, I get $10,000.

00:30:36.290 --> 00:30:37.620
I make $40,000.

00:30:37.620 --> 00:30:39.360
I lose $30,000.

00:30:39.360 --> 00:30:40.740
Let me actually write it out.

00:30:40.740 --> 00:30:42.900
If I get hit by the car,
what happens to me?

00:30:42.900 --> 00:30:46.580
Well, I make $40,000.

00:30:46.580 --> 00:30:48.070
I always make $40,000
each year.

00:30:52.640 --> 00:30:55.970
I lose $30,000, because
I get hit by the car.

00:30:55.970 --> 00:30:58.760
But then the insurance company
pays me $30,000.

00:30:58.760 --> 00:31:00.230
They pay off my debts.

00:31:00.230 --> 00:31:03.400
So then I gain $30,000.

00:31:03.400 --> 00:31:05.250
So these things cancel.

00:31:05.250 --> 00:31:07.520
But I have to pay the insurance
company premium.

00:31:07.520 --> 00:31:09.310
So I have to pay
some amount x.

00:31:12.760 --> 00:31:20.200
If I don't get hit by the car, I
get my $40,000 income, but I

00:31:20.200 --> 00:31:23.180
still have to pay the insurance
company premium.

00:31:23.180 --> 00:31:24.640
I have to pay them whether
I get hit or not.

00:31:24.640 --> 00:31:25.200
It's insurance.

00:31:25.200 --> 00:31:28.050
I pay them either way.

00:31:28.050 --> 00:31:29.800
So that's my utility.

00:31:29.800 --> 00:31:35.790
So my expected utility
with insurance is the

00:31:35.790 --> 00:31:37.040
sum of these two.

00:31:39.270 --> 00:31:42.730
And I want to set that
equal to 199.

00:31:42.730 --> 00:31:46.380
I want to say what x am I
willing to pay that would

00:31:46.380 --> 00:31:50.590
leave me at the same utility as
if I was uninsured as per

00:31:50.590 --> 00:31:52.930
the answer here?

00:31:52.930 --> 00:31:57.080
Well, it turns out that if you
do, that if you solve this,

00:31:57.080 --> 00:32:05.040
you get that x equals 399.

00:32:05.040 --> 00:32:10.870
That is you would pay $399
for insurance that has a

00:32:10.870 --> 00:32:14.350
value of only $300.

00:32:14.350 --> 00:32:18.220
You'd pay $399 for insurance
even though the actuarially

00:32:18.220 --> 00:32:20.590
fair price is $300.

00:32:20.590 --> 00:32:22.870
You would pay you insurance
company $99 more than they

00:32:22.870 --> 00:32:25.400
expect to pay out to you.

00:32:25.400 --> 00:32:25.910
Why?

00:32:25.910 --> 00:32:27.610
Because you're risk averse.

00:32:27.610 --> 00:32:31.220
Because you're made so much
sadder than being left with

00:32:31.220 --> 00:32:35.200
$10,000 than you are by
having to pay $300.

00:32:35.200 --> 00:32:36.960
If it doesn't work out,
you pay $300.

00:32:36.960 --> 00:32:37.370
Who cares?

00:32:37.370 --> 00:32:39.050
That's tiny compared
to your income.

00:32:39.050 --> 00:32:41.200
But if it does work
out, you're safe

00:32:41.200 --> 00:32:44.200
from having to starve.

00:32:44.200 --> 00:32:45.520
You pay $400, I'm sorry.

00:32:45.520 --> 00:32:47.440
You pay $399.

00:32:47.440 --> 00:32:50.380
You're like, look, I'll be
bummed if I have to pay $400.

00:32:50.380 --> 00:32:52.400
That's a percent of my
income basically.

00:32:52.400 --> 00:32:53.990
That would be a shame to pay
a percent of my income for

00:32:53.990 --> 00:32:55.210
something that doesn't happen.

00:32:55.210 --> 00:32:57.890
But, boy, would I be happy in
that 1 in 100 chance where I

00:32:57.890 --> 00:33:01.530
get hit by a car when
I'm not out $30,000.

00:33:01.530 --> 00:33:04.690
So you will pay $399
for insurance

00:33:04.690 --> 00:33:07.120
that's only worth $300.

00:33:07.120 --> 00:33:10.735
That extra $99 we call
a risk premium.

00:33:19.620 --> 00:33:22.230
We call that a risk premium.

00:33:22.230 --> 00:33:24.780
The extra $99, we call
a risk premium.

00:33:24.780 --> 00:33:30.470
That is the amount that you are
willing to pay above and

00:33:30.470 --> 00:33:34.260
beyond the fair price, because
you're risk averse.

00:33:34.260 --> 00:33:38.780
And what you should go home and
show yourself using the

00:33:38.780 --> 00:33:42.680
same kind of mathematics is
that, for example, the risk

00:33:42.680 --> 00:33:47.440
premium will rise the
bigger the loss is.

00:33:47.440 --> 00:33:48.470
Hopefully you can see the
intuition on that.

00:33:48.470 --> 00:33:51.250
The bigger the loss is for a
given level of income the

00:33:51.250 --> 00:33:52.200
bigger the risk is.

00:33:52.200 --> 00:33:54.240
Likewise, for a given
loss, the risk

00:33:54.240 --> 00:33:56.840
premium falls with income.

00:33:56.840 --> 00:33:59.430
So the bigger is the loss of
relative to income the more

00:33:59.430 --> 00:34:00.680
risk premium you're
willing to pay.

00:34:03.300 --> 00:34:06.470
You should also, obviously, see
that the more risk averse

00:34:06.470 --> 00:34:08.830
you are, the bigger premium
you're willing to pay.

00:34:08.830 --> 00:34:13.560
A risk neutral person would
not pay a risk premium.

00:34:13.560 --> 00:34:15.760
Only a risk averse
person will.

00:34:15.760 --> 00:34:17.719
So the more risk averse you are,
the bigger risk premium

00:34:17.719 --> 00:34:20.000
you'll pay, and the
bigger the loss is

00:34:20.000 --> 00:34:20.810
relative to your income.

00:34:20.810 --> 00:34:23.080
These are the same principles
we talked about before.

00:34:26.010 --> 00:34:29.120
So the $43.75 we were willing to
pay to avoid that gamble I

00:34:29.120 --> 00:34:33.050
was going to force on you, that
was the risk premium.

00:34:33.050 --> 00:34:36.300
You were willing to pay $44
to avoid that gamble.

00:34:36.300 --> 00:34:40.179
Here, you're willing to pay
$99 to avoid the risk of

00:34:40.179 --> 00:34:44.389
ending up in that bad state
where you get hit by the car.

00:34:44.389 --> 00:34:46.040
And that's why people
buy insurance.

00:34:46.040 --> 00:34:49.239
And that's why insurance
companies make

00:34:49.239 --> 00:34:51.130
ungodly amounts of money.

00:34:51.130 --> 00:34:53.250
In the US we have a health
insurance industry, for

00:34:53.250 --> 00:34:58.620
example, that earns about
$800 billion a year.

00:34:58.620 --> 00:34:59.830
Why do they make
all that money?

00:34:59.830 --> 00:35:01.600
Because people are risk averse,
and they're willing to

00:35:01.600 --> 00:35:04.130
pay to have someone else
bear the risk of

00:35:04.130 --> 00:35:07.370
their injury or illness.

00:35:07.370 --> 00:35:09.770
Any questions about that?

00:35:09.770 --> 00:35:12.180
Now, I don't mean by that to
say, insurance is a bad thing,

00:35:12.180 --> 00:35:13.360
and we shouldn't do it.

00:35:13.360 --> 00:35:15.310
Risk aversion is the nature
of our utility functions.

00:35:15.310 --> 00:35:17.260
We should be willing to
pay a risk premium.

00:35:17.260 --> 00:35:19.170
It's just that you need to
understand why, in fact, it

00:35:19.170 --> 00:35:23.030
makes sense to have insurance
in that case.

00:35:23.030 --> 00:35:25.200
The second application
is the lottery.

00:35:32.300 --> 00:35:35.460
The lottery is a total ripoff.

00:35:35.460 --> 00:35:37.005
I hope you knew this already.

00:35:37.005 --> 00:35:39.580
The expected value
of $1 lottery

00:35:39.580 --> 00:35:42.230
ticket is roughly $0.50.

00:35:42.230 --> 00:35:44.860
So for every $1 you spend in the
lottery, in expectation,

00:35:44.860 --> 00:35:46.870
you get about $0.50 back.

00:35:46.870 --> 00:35:52.930
This is an incredibly bad bet,
incredibly unfair, an

00:35:52.930 --> 00:35:55.160
incredibly unfair bet.

00:35:55.160 --> 00:35:59.930
On average, you lose $0.50
for every $1 you bet.

00:35:59.930 --> 00:36:04.700
So, basically, despite that,
lotteries are wildly popular.

00:36:04.700 --> 00:36:06.790
They've become a huge source
of revenue for state

00:36:06.790 --> 00:36:07.820
governments.

00:36:07.820 --> 00:36:09.900
A lot of the money that state
governments now take in is

00:36:09.900 --> 00:36:12.120
through state lotteries.

00:36:12.120 --> 00:36:16.860
What accounts for the fact that
lotteries are so popular?

00:36:16.860 --> 00:36:19.900
Well, there's four different
theories for why lotteries are

00:36:19.900 --> 00:36:21.510
so popular.

00:36:21.510 --> 00:36:25.030
The first is that people
are risk-loving.

00:36:25.030 --> 00:36:27.120
We have it all wrong.

00:36:27.120 --> 00:36:28.560
Actually people like taking
risks, and the

00:36:28.560 --> 00:36:29.810
lottery feeds that.

00:36:32.300 --> 00:36:34.110
This, of course, we can
immediately rule out.

00:36:34.110 --> 00:36:34.990
How?

00:36:34.990 --> 00:36:37.920
How do we know this is wrong?

00:36:37.920 --> 00:36:39.200
That the answer is that people
play the lottery because

00:36:39.200 --> 00:36:40.195
they're risk-loving.

00:36:40.195 --> 00:36:42.512
How do we know people
aren't risk-loving?

00:36:42.512 --> 00:36:44.380
AUDIENCE: The same people don't
take [UNINTELLIGIBLE]

00:36:44.380 --> 00:36:45.957
PROFESSOR: And they spend
$800 billion a

00:36:45.957 --> 00:36:47.930
year on health insurance.

00:36:47.930 --> 00:36:50.947
Basically, as a society, we
spend, in total, about $1.5

00:36:50.947 --> 00:36:54.000
trillion a year on insuring
various risks that face us.

00:36:54.000 --> 00:36:56.080
We're not risk-loving.

00:36:56.080 --> 00:36:59.620
So that's clearly
not the answer.

00:36:59.620 --> 00:37:01.840
However, there's
an alternative.

00:37:01.840 --> 00:37:09.280
People could basically alternate
between risk-loving

00:37:09.280 --> 00:37:10.530
and risk-aversion.

00:37:12.460 --> 00:37:15.230
This is a theory due to Milton
Friedman, the famous economist

00:37:15.230 --> 00:37:17.680
from Chicago and a co-author
named Savage, the

00:37:17.680 --> 00:37:20.960
Friedman-Savage preferences,
where the notion is that

00:37:20.960 --> 00:37:27.030
basically people are risk averse
over small gambles but

00:37:27.030 --> 00:37:29.060
risk-loving over
large gambles.

00:37:29.060 --> 00:37:31.980
So to see that, go last
figure in the graph.

00:37:31.980 --> 00:37:34.580
This is sort of a complicated
case.

00:37:34.580 --> 00:37:39.000
Basically, the notion is if you
take someone, they have a

00:37:39.000 --> 00:37:42.280
utility function which is
initially risk averse and then

00:37:42.280 --> 00:37:43.470
becomes risk-loving.

00:37:43.470 --> 00:37:48.040
That is in the segment between
W1 and W3, that looks like a

00:37:48.040 --> 00:37:50.140
risk averse utility function.

00:37:50.140 --> 00:37:53.060
But once you get above W3, it
looks like a risk-loving

00:37:53.060 --> 00:37:54.940
utility function.

00:37:54.940 --> 00:38:01.550
So the notion is that for things
which can make me very

00:38:01.550 --> 00:38:04.060
poor, I'm risk averse.

00:38:04.060 --> 00:38:08.230
I want to insure against events
which will leave me in

00:38:08.230 --> 00:38:09.380
that bottom segment.

00:38:09.380 --> 00:38:13.910
But once I'm going to be
above W3, then great.

00:38:13.910 --> 00:38:14.880
I'm happy to take risks.

00:38:14.880 --> 00:38:16.130
Then I become risk-loving.

00:38:18.160 --> 00:38:24.900
Now, this is a not crazy idea.

00:38:24.900 --> 00:38:28.380
Graphically, what I'm showing
you here, is that b* is

00:38:28.380 --> 00:38:30.610
utility without the gamble
and b is with.

00:38:30.610 --> 00:38:32.740
So you see you're happier
without the gamble when your

00:38:32.740 --> 00:38:33.820
income is low.

00:38:33.820 --> 00:38:36.380
Once your income is a lot
higher, you're happier with

00:38:36.380 --> 00:38:40.510
the gamble at d then you are
without the gamble at d*.

00:38:40.510 --> 00:38:42.550
That's not a crazy theory.

00:38:42.550 --> 00:38:44.860
The notion is that once I'm
rich enough, I become

00:38:44.860 --> 00:38:46.110
risk-loving.

00:38:46.110 --> 00:38:49.840
But when I'm poor, I don't
want to take the risks.

00:38:49.840 --> 00:38:52.750
The problem is that this is
inconsistent with lottery

00:38:52.750 --> 00:38:54.400
behavior in the following
sense.

00:38:54.400 --> 00:38:57.200
Most people who play
the lottery don't

00:38:57.200 --> 00:38:58.740
play the Mega Millions.

00:38:58.740 --> 00:39:01.035
They play tiny scratch
lotteries where you

00:39:01.035 --> 00:39:04.730
bet $1 to win $10.

00:39:04.730 --> 00:39:07.580
And people spend huge amounts
of money on lotteries with

00:39:07.580 --> 00:39:10.000
very, very low payoffs.

00:39:10.000 --> 00:39:12.950
That is inconsistent
with this.

00:39:12.950 --> 00:39:14.826
Because this would say that
you'd only play lotteries that

00:39:14.826 --> 00:39:15.970
have big payoffs.

00:39:15.970 --> 00:39:18.520
Lotteries that have small
payoffs, once again, there's

00:39:18.520 --> 00:39:21.460
no reason to play that and
still buy insurance.

00:39:21.460 --> 00:39:24.550
So if you're buying insurance
against being low income, why

00:39:24.550 --> 00:39:26.190
are you playing these small
lotteries that are a ripoff.

00:39:26.190 --> 00:39:28.290
Because those small ones
are a ripoff too.

00:39:28.290 --> 00:39:30.340
So the existence of the fact
that the most popular

00:39:30.340 --> 00:39:32.350
lotteries are actually the
small lotteries is

00:39:32.350 --> 00:39:34.200
inconsistent with this
explanation.

00:39:34.200 --> 00:39:35.306
Yeah.

00:39:35.306 --> 00:39:36.680
AUDIENCE: So I'm confused.

00:39:36.680 --> 00:39:39.430
Is it risk-loving on
large gambles?

00:39:39.430 --> 00:39:40.695
PROFESSOR: Yeah, risk-loving
on large gambles.

00:39:44.860 --> 00:39:46.390
It's not the size
of the gamble.

00:39:46.390 --> 00:39:48.760
You're risk-loving on gambles
which leave you in a high

00:39:48.760 --> 00:39:50.890
wealth state.

00:39:50.890 --> 00:39:53.220
The point is that if
I'm gambling over

00:39:53.220 --> 00:39:55.035
winning Mega Millions.

00:39:55.035 --> 00:39:56.320
Yeah, I'm a little
risk averse.

00:39:56.320 --> 00:39:59.000
But the truth is winning Mega
Millions would make me so

00:39:59.000 --> 00:40:00.770
happy that I could move into
the risk-loving part of my

00:40:00.770 --> 00:40:02.310
utility function.

00:40:02.310 --> 00:40:04.940
But this would not explain why
people ever play something

00:40:04.940 --> 00:40:07.470
that pays off $100.

00:40:07.470 --> 00:40:10.190
This is a fancy way of the
intuition you probably have.

00:40:10.190 --> 00:40:10.900
It's I'd think differently about
something which would

00:40:10.900 --> 00:40:13.010
completely change my life and
make me a multi-billionaire,

00:40:13.010 --> 00:40:15.730
that's something that would make
me raise me, than the bet

00:40:15.730 --> 00:40:17.560
I offered you guys before.

00:40:17.560 --> 00:40:19.590
People are systematically taking
terrible bets like the

00:40:19.590 --> 00:40:21.200
kind i offered you
guys before.

00:40:21.200 --> 00:40:24.725
And that's inconsistent with
these preferences.

00:40:24.725 --> 00:40:28.410
The third explanation
is entertainment.

00:40:31.360 --> 00:40:35.110
It's that the utility function
has in it the

00:40:35.110 --> 00:40:37.510
thrill of the risk.

00:40:37.510 --> 00:40:39.600
We only write down utility
functions that are a function

00:40:39.600 --> 00:40:41.980
of consumption like how many
pizza and movies you see.

00:40:41.980 --> 00:40:44.780
But people have utility
over lots of things.

00:40:44.780 --> 00:40:46.770
One thing you may have utility
of the thrill of being able to

00:40:46.770 --> 00:40:50.010
scratch the thing off and seeing
if they won or not.

00:40:50.010 --> 00:40:53.580
That would actually be
consistent with the fact that

00:40:53.580 --> 00:40:55.100
people play a lot of
small lotteries.

00:40:55.100 --> 00:40:57.540
If it's a thrill of winning
that matters, if it's the

00:40:57.540 --> 00:40:59.560
scratch off thrill that matters,
then the optimal

00:40:59.560 --> 00:41:02.480
thing to do, in fact, would be
to not play one Mega Million.

00:41:02.480 --> 00:41:04.530
It would be to play lots
of little lotteries.

00:41:04.530 --> 00:41:06.490
And that would be consistent
with that behavior.

00:41:06.490 --> 00:41:08.550
So one story that is consistent
with what we see is

00:41:08.550 --> 00:41:10.930
that people actually view
this as entertainment.

00:41:10.930 --> 00:41:12.450
On the other hand, once again,
it's really expensive

00:41:12.450 --> 00:41:13.220
entertainment.

00:41:13.220 --> 00:41:17.380
Because you're throwing away
$0.50 of every $1.

00:41:17.380 --> 00:41:18.950
So you've got to get a lot of
enjoyment out of that scratch

00:41:18.950 --> 00:41:22.060
off relative to when you
go to see a movie.

00:41:22.060 --> 00:41:23.310
So that's another theory.

00:41:36.380 --> 00:41:37.260
I'm going to put this in here.

00:41:37.260 --> 00:41:39.080
It sort of inserts in here.

00:41:39.080 --> 00:41:42.330
We talked about the fact that
people can't be risk-loving

00:41:42.330 --> 00:41:45.220
because they buy insurance.

00:41:45.220 --> 00:41:46.840
And this alternating thing
doesn't work, because they

00:41:46.840 --> 00:41:48.500
play small lotteries.

00:41:48.500 --> 00:41:51.650
But another theory that might
fit here is a theory we call

00:41:51.650 --> 00:41:53.400
loss aversion.

00:41:53.400 --> 00:41:55.990
This is sort of a different
version of the Friedman-Savage

00:41:55.990 --> 00:41:58.100
preferences.

00:41:58.100 --> 00:42:01.235
It's that people are, in
general, risk averse.

00:42:03.850 --> 00:42:08.120
But, in fact, they're really
risk averse on the downside,

00:42:08.120 --> 00:42:10.850
and they don't care so
much on the upside.

00:42:10.850 --> 00:42:14.200
So, in other words, the point
is that when I initially

00:42:14.200 --> 00:42:20.040
offered you that bet of win
$125, lose $100, part of your

00:42:20.040 --> 00:42:21.950
reaction was about the
risk aversion.

00:42:21.950 --> 00:42:23.470
But a lot of you are thinking,
I'd be really

00:42:23.470 --> 00:42:25.170
bummed if I lost $100.

00:42:25.170 --> 00:42:26.160
It's not just that I don't
have it to spare.

00:42:26.160 --> 00:42:28.700
It's just like, god, I
would kick myself.

00:42:28.700 --> 00:42:29.890
It was one flip of the coin.

00:42:29.890 --> 00:42:32.990
How could I possibly have
been so stupid?

00:42:32.990 --> 00:42:34.330
Whereas if you won,
you'd be happy.

00:42:34.330 --> 00:42:37.010
But then you'd go on
to the next class.

00:42:37.010 --> 00:42:39.860
The notion is that basically
it's an extreme version of

00:42:39.860 --> 00:42:41.480
risk aversion.

00:42:41.480 --> 00:42:42.850
It's not only that you're
risk averse, it go

00:42:42.850 --> 00:42:43.760
further than that.

00:42:43.760 --> 00:42:47.150
Relative to the starting point,
anything which is a

00:42:47.150 --> 00:42:49.670
loss really pisses you off.

00:42:49.670 --> 00:42:53.860
So, in fact, even that little
gamble I offered you, win

00:42:53.860 --> 00:42:56.560
$1.25 lose $1, you still
might not take.

00:42:56.560 --> 00:42:58.850
Some of you still wouldn't
take it.

00:42:58.850 --> 00:43:00.130
And the reason you
wouldn't take it

00:43:00.130 --> 00:43:01.635
can't be risk aversion.

00:43:01.635 --> 00:43:03.680
Because it's just too small
for risk aversion

00:43:03.680 --> 00:43:04.270
to plausibly work.

00:43:04.270 --> 00:43:06.505
It's that you'll just be bummed
that you did that and

00:43:06.505 --> 00:43:07.710
you took that chance.

00:43:07.710 --> 00:43:10.520
You'd be made sadder by the
loss than you'd be made

00:43:10.520 --> 00:43:13.520
happier by the win.

00:43:13.520 --> 00:43:17.740
In that case, that could explain
why people spend a lot

00:43:17.740 --> 00:43:19.110
of money to buy insurance.

00:43:19.110 --> 00:43:23.100
Because they'll be so bummed
if things go badly.

00:43:23.100 --> 00:43:28.230
But they might play the lottery
because, in fact,

00:43:28.230 --> 00:43:30.760
around that point, they don't
view the money they're

00:43:30.760 --> 00:43:32.040
spending as a loss.

00:43:32.040 --> 00:43:32.710
They think of it differently.

00:43:32.710 --> 00:43:36.280
They think of the loss of being
my house burned down.

00:43:36.280 --> 00:43:36.910
That's a loss.

00:43:36.910 --> 00:43:37.650
That would make me really sad.

00:43:37.650 --> 00:43:39.860
But the $1 I paid to pay
the lottery, that's

00:43:39.860 --> 00:43:41.860
not really a loss.

00:43:41.860 --> 00:43:45.520
So I'm risk neutral going
up and really risk

00:43:45.520 --> 00:43:46.910
averse going down.

00:43:46.910 --> 00:43:49.330
So I'm willing to take gambles
that push me up.

00:43:49.330 --> 00:43:52.012
It's sort of like
Friedman-Savage.

00:43:52.012 --> 00:43:54.060
I'm willling to take gambles
that push me up, not gambles

00:43:54.060 --> 00:43:54.660
that pull me down.

00:43:54.660 --> 00:43:58.430
But, once again, that doesn't
really explain the small ones.

00:43:58.430 --> 00:44:00.040
That doesn't really explain
the small ones.

00:44:00.040 --> 00:44:02.380
That's more the entertainment
theory.

00:44:02.380 --> 00:44:05.920
Then finally, the last
theory we have is

00:44:05.920 --> 00:44:07.170
that people are stupid.

00:44:09.700 --> 00:44:13.750
The lottery is, after all,
its official motto is

00:44:13.750 --> 00:44:15.890
a tax on the stupid.

00:44:15.890 --> 00:44:16.710
And that's what it is.

00:44:16.710 --> 00:44:17.710
It's a tax on the stupid.

00:44:17.710 --> 00:44:21.080
Basically many of your public
schools are financed by taxes

00:44:21.080 --> 00:44:21.980
paid by stupid people.

00:44:21.980 --> 00:44:23.930
It's sort of ironic.

00:44:23.930 --> 00:44:27.350
But people just don't know.

00:44:27.350 --> 00:44:30.180
You probably all had a vague
sense that the lottery wasn't

00:44:30.180 --> 00:44:31.350
a sensible thing to play.

00:44:31.350 --> 00:44:32.700
But how many people actually
knew it was that

00:44:32.700 --> 00:44:33.710
bad a deal as I said.

00:44:33.710 --> 00:44:36.920
That is actually was $0.50
expected payoff.

00:44:36.920 --> 00:44:38.160
A few of you knew.

00:44:38.160 --> 00:44:41.070
But most of you knewm had a
vague sense it was a bad deal.

00:44:41.070 --> 00:44:42.350
You didn't know how
bad a deal it was.

00:44:42.350 --> 00:44:43.840
This is sort of hard
to figure out.

00:44:43.840 --> 00:44:46.920
Meanwhile, you see on TV that
these guys win these bazillion

00:44:46.920 --> 00:44:50.060
dollars, and you get the thrill
of scratching if off.

00:44:50.060 --> 00:44:55.035
So, basically, if people are
just stupid, then that could

00:44:55.035 --> 00:44:55.300
explain it.

00:44:55.300 --> 00:44:58.550
The problem is it matters a
lot for government policy

00:44:58.550 --> 00:45:00.640
which of these is right.

00:45:00.640 --> 00:45:03.140
Because if A through C is right,
if one through three

00:45:03.140 --> 00:45:06.780
are right, then the government
should go

00:45:06.780 --> 00:45:07.880
ahead and allow lotteries.

00:45:07.880 --> 00:45:10.490
And there's no reason why the
state shouldn't run a lottery.

00:45:10.490 --> 00:45:12.170
In fact, let's take the
entertainment theory.

00:45:12.170 --> 00:45:14.260
If this is really entertainment,
and the state

00:45:14.260 --> 00:45:15.800
can make money off of
my entertainment,

00:45:15.800 --> 00:45:17.920
then that's a win-win.

00:45:17.920 --> 00:45:19.750
I'm happy, because I'm
playing the lottery.

00:45:19.750 --> 00:45:21.960
The state is happy, because
it's financing schools.

00:45:21.960 --> 00:45:23.220
That's a win-win.

00:45:23.220 --> 00:45:27.420
So if these are right, you're
going to want to encourage

00:45:27.420 --> 00:45:29.090
state lotteries.

00:45:29.090 --> 00:45:31.760
But if this one's right, we
don't want to have them.

00:45:31.760 --> 00:45:35.365
Because, A terrible way to raise
government revenues is

00:45:35.365 --> 00:45:37.740
to tax stupid people.

00:45:37.740 --> 00:45:39.310
There are much better ways to
raise government revenues.

00:45:39.310 --> 00:45:41.240
We'll talk about taxation
in a couple of lectures.

00:45:41.240 --> 00:45:43.270
But, clearly, taxing the stupid
is not going to be an

00:45:43.270 --> 00:45:43.980
optimal tax.

00:45:43.980 --> 00:45:45.018
Yeah.

00:45:45.018 --> 00:45:47.826
AUDIENCE: I can maybe sort of
understand why people would

00:45:47.826 --> 00:45:50.166
prefer smaller lotteries
over bigger lotteries.

00:45:50.166 --> 00:45:53.050
Because they are thinking that
in smaller lotteries, they

00:45:53.050 --> 00:45:54.953
have a much bigger
chance of winning

00:45:54.953 --> 00:45:55.780
than in bigger lotteries.

00:45:55.780 --> 00:46:00.785
So, in that sense, their
expected payoff in terms of

00:46:00.785 --> 00:46:03.719
utility or other
[INAUDIBLE PHRASE]

00:46:03.719 --> 00:46:07.435
is a lot higher than the antes
in the bigger ones, even

00:46:07.435 --> 00:46:11.450
though the bigger ones might end
up being a lot heavier--

00:46:11.450 --> 00:46:13.100
PROFESSOR: So that's sort of an
entertainment theory, which

00:46:13.100 --> 00:46:16.260
is my utility derives
from the win.

00:46:16.260 --> 00:46:19.430
You have a theory in mind my
utility derives from the win.

00:46:19.430 --> 00:46:21.710
Because if it's just
about dollars, that

00:46:21.710 --> 00:46:22.430
wouldn't explain it.

00:46:22.430 --> 00:46:25.080
Because I win so many more from
the big one that it would

00:46:25.080 --> 00:46:26.870
compensate from the frequency
at which I'd

00:46:26.870 --> 00:46:27.760
win the little one.

00:46:27.760 --> 00:46:30.430
But if I actually, in my utility
function, have the joy

00:46:30.430 --> 00:46:33.240
of seeing that winning thing,
then that would explain it.

00:46:33.240 --> 00:46:34.540
That's an entertainment
theory.

00:46:34.540 --> 00:46:36.790
You're saying, in my utility
function, I actually get joy

00:46:36.790 --> 00:46:38.430
from scratching off and seeing
that it's a winner, and so

00:46:38.430 --> 00:46:43.070
much joy that I'd much rather
take a 10% chance at a small

00:46:43.070 --> 00:46:45.180
win than a 1% chance
at a huge win.

00:46:45.180 --> 00:46:47.800
Because then, at least, with the
first one, 1 in 10 times I

00:46:47.800 --> 00:46:49.990
get that joy of the scratch
off and seeing it's a win.

00:46:49.990 --> 00:46:51.300
So that's sort of
an explanation.

00:46:51.300 --> 00:46:55.820
And that would say that
lotteries are good.

00:46:55.820 --> 00:46:57.550
The other way economists might
think about lotteries is

00:46:57.550 --> 00:46:59.320
they're voluntary taxes.

00:46:59.320 --> 00:47:02.000
The public doesn't like taxes.

00:47:02.000 --> 00:47:04.870
Here's a voluntary tax.

00:47:04.870 --> 00:47:08.530
You never hear policy makers
getting up and railing against

00:47:08.530 --> 00:47:11.280
a horrible evils
of the lottery.

00:47:11.280 --> 00:47:13.000
Sometimes groups do.

00:47:13.000 --> 00:47:15.670
Sometimes outside groups
do and stuff.

00:47:15.670 --> 00:47:16.760
But politicians don't.

00:47:16.760 --> 00:47:18.940
But those same politicians will
go on and on about how

00:47:18.940 --> 00:47:19.860
terrible taxes are.

00:47:19.860 --> 00:47:20.220
I'm going to cut your taxes.

00:47:20.220 --> 00:47:21.875
Taxes are terrible.

00:47:21.875 --> 00:47:24.460
Well, the lottery is a voluntary
tax in that sense.

00:47:24.460 --> 00:47:26.210
And I might say, look, there's
no reason to oppose it, it's a

00:47:26.210 --> 00:47:27.130
voluntary tax.

00:47:27.130 --> 00:47:28.200
It's those involuntary
taxes that

00:47:28.200 --> 00:47:31.180
cause problems in society.

00:47:31.180 --> 00:47:34.080
Well, whether we want to buy
that story or not depends on

00:47:34.080 --> 00:47:36.710
how much we think it's being
played because people are

00:47:36.710 --> 00:47:37.305
stupid or not.

00:47:37.305 --> 00:47:37.560
OK.

00:47:37.560 --> 00:47:39.120
Let me stop there.

00:47:39.120 --> 00:47:41.500
So that's a great example of
how a little bit of an

00:47:41.500 --> 00:47:44.760
extension of our model
can really enrich our

00:47:44.760 --> 00:47:47.070
understanding about a lot of
decisions that we make in the

00:47:47.070 --> 00:47:48.000
real world.

00:47:48.000 --> 00:47:49.175
We'll come back and
talk about another

00:47:49.175 --> 00:47:53.500
version like that later.

00:47:53.500 --> 00:47:55.650
And that is the case of
thinking about savings

00:47:55.650 --> 00:47:58.480
decisions and thinking about
individual decisions on how

00:47:58.480 --> 00:47:59.980
much to save and how
much to spend.