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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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00:00:48,060 --> 00:00:49,000
this person.

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00:00:49,000 --> 00:00:51,010
I've got uncertainty about
whether I should refinance my

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00:00:51,010 --> 00:00:54,100
mortgage, or which college to
send my kid to, or how much

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00:00:54,100 --> 00:00:56,140
life insurance I should buy.

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00:00:56,140 --> 00:00:58,850
Uncertainty only get more
and more important as

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you move on in life.

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00:00:59,610 --> 00:01:01,070
This is an important issue.

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00:01:01,070 --> 00:01:02,900
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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00:01:07,740 --> 00:01:10,500
once again, to make simplifying
assumptions which

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00:01:10,500 --> 00:01:12,760
allow us to write down sensible
models, but which

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00:01:12,760 --> 00:01:15,220
capture the key elements of
what we're thinking about.

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00:01:15,220 --> 00:01:17,520
And the simplifying assumption
here is we move to the tools

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00:01:17,520 --> 00:01:19,500
of what we call expected
utility theory.

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00:01:29,130 --> 00:01:33,790
And so, basically, the way we
think about expected utility

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00:01:33,790 --> 00:01:34,590
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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00:01:40,610 --> 00:01:42,130
answer to this.

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00:01:42,130 --> 00:01:43,890
But I do want you guys
to answer me.

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00:01:43,890 --> 00:01:44,580
There's no right answer.

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00:01:44,580 --> 00:01:45,250
Here's the question.

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00:01:45,250 --> 00:01:46,320
I'm going to give
you a choice.

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00:01:46,320 --> 00:01:47,190
I'm going to flip a coin.

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00:01:47,190 --> 00:01:49,120
I have a coin in pocket, and
I'm going to flip it.

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00:01:49,120 --> 00:01:50,260
And I'm going to offer
you guys the

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00:01:50,260 --> 00:01:51,670
ability to make a bet.

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00:01:51,670 --> 00:01:56,020
If it comes up heads,
you win $125.

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00:01:56,020 --> 00:01:59,080
If it comes up tails,
you lose $100.

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00:01:59,080 --> 00:02:00,450
Heads, you win a $125.

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00:02:00,450 --> 00:02:02,270
Tails, you lose $100.

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00:02:02,270 --> 00:02:03,470
There's no right answer.

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How many would take that bet.

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00:02:07,440 --> 00:02:08,800
How many people would
not take that bet?

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00:02:11,580 --> 00:02:12,120
Very good.

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00:02:12,120 --> 00:02:14,770
That's the typical set of
responses I get to this.

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00:02:14,770 --> 00:02:17,210
Now, what's interesting
is to think about the

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00:02:17,210 --> 00:02:18,350
parameters of that bet.

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00:02:18,350 --> 00:02:20,090
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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00:02:22,410 --> 00:02:23,660
expected value.

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What's the expected value
of that gamble?

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00:02:30,100 --> 00:02:33,140
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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00:03:19,680 --> 00:03:22,180
$12.50 per time.

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00:03:22,180 --> 00:03:24,000
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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00:03:50,311 --> 00:03:52,610
Yet, the majority of you
would not be willing

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00:03:52,610 --> 00:03:55,050
to take this bet.

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00:03:55,050 --> 00:03:56,350
In fact, the majority
of people would

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00:03:56,350 --> 00:03:57,540
not take this bet.

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00:03:57,540 --> 00:04:00,480
Why is that?

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00:04:00,480 --> 00:04:03,100
Why is it that I've dictated
a bet which has a positive

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00:04:03,100 --> 00:04:04,750
expected value and yet,
people won't take it.

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00:04:04,750 --> 00:04:05,120
Yeah.

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00:04:05,120 --> 00:04:07,880
AUDIENCE: But wouldn't that also
depend on how much money

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00:04:07,880 --> 00:04:08,340
you have.

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PROFESSOR: It will absolutely
depend on how much money you

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00:04:09,880 --> 00:04:10,100
have.

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00:04:10,100 --> 00:04:10,573
AUDIENCE: Right.

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00:04:10,573 --> 00:04:14,357
So if I were a richer person,
then losing $100 isn't as

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00:04:14,357 --> 00:04:17,680
important to me as the chance
of getting $125.

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00:04:17,680 --> 00:04:17,940
PROFESSOR: OK.

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00:04:17,940 --> 00:04:18,790
So flesh that out.

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00:04:18,790 --> 00:04:20,110
Why is that?

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00:04:20,110 --> 00:04:23,280
Why is it that basically it
would matter how much wealth

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00:04:23,280 --> 00:04:25,430
you have. Because no matter how
much wealth you have, this

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00:04:25,430 --> 00:04:27,380
math is impeachable.

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00:04:27,380 --> 00:04:29,370
It's always a good bet.

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00:04:29,370 --> 00:04:32,770
So why is it that your state
without much wealth, your

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00:04:32,770 --> 00:04:34,940
state as college students
without much wealth, what is

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00:04:34,940 --> 00:04:37,270
it about you that causes you to
not want to take this bet

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00:04:37,270 --> 00:04:38,520
that's more than fair.

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00:04:40,710 --> 00:04:46,960
AUDIENCE: So, basically, for me,
the risk of losing or the

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00:04:46,960 --> 00:04:50,197
state I will be in after I lose
is much greater, well,

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00:04:50,197 --> 00:04:53,940
for me, a lot more than what
I would be in if win.

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00:04:53,940 --> 00:04:54,850
PROFESSOR: Exactly.

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00:04:54,850 --> 00:04:57,130
And there's two possible
reasons for that.

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00:04:57,130 --> 00:05:00,310
One we're going to push off to
the very end of the lecture.

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00:05:00,310 --> 00:05:02,880
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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00:05:05,970 --> 00:05:08,950
expected utility, and
individuals are risk averse.

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00:05:13,500 --> 00:05:17,450
Expected utility is going to
differ from expected value

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00:05:17,450 --> 00:05:19,880
when individuals are
risk averse.

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00:05:19,880 --> 00:05:22,320
Expecting utility is not going
to be the probability times

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00:05:22,320 --> 00:05:23,640
the value if you lose.

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00:05:23,640 --> 00:05:26,790
Expected utility is going to
be the probability that you

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00:05:26,790 --> 00:05:32,200
lose times the utility if you
lose plus the probability that

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00:05:32,200 --> 00:05:36,190
you win times the utility
if you win.

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00:05:36,190 --> 00:05:40,280
And utility is not the same as
value, importantly, because

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utility functions exhibit
diminishing marginal utility.

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00:05:44,950 --> 00:05:47,065
Utility functions
are not linear.

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00:05:47,065 --> 00:05:47,950
Utility functions
are nonlinear.

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00:05:47,950 --> 00:05:50,970
And, in particular, there's
diminishing marginal utility.

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00:05:50,970 --> 00:05:57,310
And with diminishing marginal
utility, you're going to not

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00:05:57,310 --> 00:06:04,145
want bets where there's the
chance you lose is equal to or

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00:06:04,145 --> 00:06:06,650
even a bit smaller than the
value that you win.

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00:06:06,650 --> 00:06:10,870
And the basic point is that the
joy of winning is smaller

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00:06:10,870 --> 00:06:14,060
than the pain of losing with
diminishing marginal utility.

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00:06:14,060 --> 00:06:14,558
Yeah.

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00:06:14,558 --> 00:06:18,542
AUDIENCE: Isn't there also a
statistical side to this then?

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00:06:18,542 --> 00:06:22,028
Because we don't know how many
times we're going to bet.

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00:06:22,028 --> 00:06:23,522
It might just be once.

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00:06:23,522 --> 00:06:26,708
We're a lot more comfortable if,
let's say, use the law of

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00:06:26,708 --> 00:06:29,332
large numbers and say, OK, it's
going to eventually even

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00:06:29,332 --> 00:06:30,992
out so we'll win
$12.50 a game.

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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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00:06:34,277 --> 00:06:36,990
really unlucky and flip eight
tails and two heads.

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00:06:36,990 --> 00:06:38,470
PROFESSOR: But, once again, if
you weren't risk averse, you

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00:06:38,470 --> 00:06:39,840
wouldn't care about that.

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00:06:39,840 --> 00:06:41,220
Hold that thought.

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00:06:41,220 --> 00:06:43,650
I'm going to explain why
that isn't true.

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00:06:43,650 --> 00:06:45,320
So just hold that thought.

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00:06:45,320 --> 00:06:48,740
So now let's imagine that your
utility functions are the

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00:06:48,740 --> 00:06:50,750
typical form we've worked
with before, the typical

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00:06:50,750 --> 00:06:53,280
diminishing marginal utility
form we've worked with before

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00:06:53,280 --> 00:06:56,130
where utility is the square
root of consumption.

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00:06:56,130 --> 00:06:58,940
You're casting your mind back
to consumer theory here.

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00:06:58,940 --> 00:07:00,640
You're going to have to start
integrating the course now,

156
00:07:00,640 --> 00:07:02,300
both consumer and
producer theory.

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00:07:02,300 --> 00:07:04,700
So remember we said the typical
diminishing marginal

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00:07:04,700 --> 00:07:08,430
utility function we worked
with was u equals the

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00:07:08,430 --> 00:07:09,680
square root of c.

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00:07:12,340 --> 00:07:16,230
Now, let's say you start with
consumption of $100.

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00:07:16,230 --> 00:07:17,220
Imagine you consume
your income.

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00:07:17,220 --> 00:07:19,560
Let's say you have consumption
of $100.

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00:07:19,560 --> 00:07:22,190
Well, then utility is 10.

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00:07:22,190 --> 00:07:24,130
If you start with a consumption
of $100, your

165
00:07:24,130 --> 00:07:26,530
utility is 10.

166
00:07:26,530 --> 00:07:28,090
Now let's calculate
the expected

167
00:07:28,090 --> 00:07:30,560
utility of this gamble.

168
00:07:30,560 --> 00:07:34,090
The expected utility of this
gamble is that there's a 50%

169
00:07:34,090 --> 00:07:37,240
chance that you lose.

170
00:07:37,240 --> 00:07:39,250
And, if you lose, what
is your utility?

171
00:07:39,250 --> 00:07:40,860
Well you lose $100.

172
00:07:40,860 --> 00:07:43,230
So consumption goes to 0.

173
00:07:43,230 --> 00:07:49,990
So utility is 0 plus a 50%
chance that you win.

174
00:07:49,990 --> 00:07:51,390
Well, what do you
get if you win.

175
00:07:51,390 --> 00:07:54,510
Well, if you win, you go
from $100 to $225.

176
00:07:54,510 --> 00:07:59,600
So your utility is the square
root of $225, or $15.

177
00:07:59,600 --> 00:08:04,720
Your utility is the square
root of $225 or 15.

178
00:08:04,720 --> 00:08:08,590
It's half chance of having 15.

179
00:08:08,590 --> 00:08:09,000
I'm sorry.

180
00:08:09,000 --> 00:08:10,780
So this is a negative.

181
00:08:10,780 --> 00:08:11,480
Yeah.

182
00:08:11,480 --> 00:08:14,330
So utility, if you take this
gamble, is you end up with a

183
00:08:14,330 --> 00:08:16,810
utility of 7.5.

184
00:08:16,810 --> 00:08:18,000
So utility falls.

185
00:08:18,000 --> 00:08:20,960
You move from a utility of 10
without the gamble to a

186
00:08:20,960 --> 00:08:23,730
utility of 7.5 with
the gamble.

187
00:08:23,730 --> 00:08:26,520
Utility is lower with the
gamble, which is why people

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00:08:26,520 --> 00:08:29,080
decided they didn't want
to take that gamble.

189
00:08:29,080 --> 00:08:31,630
Utility is lower.

190
00:08:31,630 --> 00:08:34,789
And the reason is because given
a utility function of

191
00:08:34,789 --> 00:08:38,840
this form, you are sadder about
losing than happier

192
00:08:38,840 --> 00:08:41,530
about winning.

193
00:08:41,530 --> 00:08:46,680
To see that, we can see that
graphically in Figure 20-1.

194
00:08:46,680 --> 00:08:50,510
This graph's utility against
wealth-- we don't usually

195
00:08:50,510 --> 00:08:52,370
graph utility, because
it's not cardinal.

196
00:08:52,370 --> 00:08:53,110
Remember it's just ordinal.

197
00:08:53,110 --> 00:08:55,050
But the sort of gives you a
sense of the intuition.

198
00:08:55,050 --> 00:08:59,020
This is a graph of utility
against wealth levels.

199
00:08:59,020 --> 00:09:06,710
So you start at point A. You
start with $100 in wealth,

200
00:09:06,710 --> 00:09:08,000
which is consumption.

201
00:09:08,000 --> 00:09:10,390
and utility of 10.

202
00:09:10,390 --> 00:09:14,110
Now, I give you a choice
of a gamble.

203
00:09:14,110 --> 00:09:18,700
That gamble has a 50% chance of
leaving you at 0 and a 50%

204
00:09:18,700 --> 00:09:23,570
chance a leaving you at point
B. So your utility and

205
00:09:23,570 --> 00:09:26,950
expected value is the midpoint
of that chord that runs from 0

206
00:09:26,950 --> 00:09:32,530
to B or point C. Your expecting
utility is lower

207
00:09:32,530 --> 00:09:35,370
than your initial utility.

208
00:09:35,370 --> 00:09:35,830
Why?

209
00:09:35,830 --> 00:09:40,190
Because utility is concave. You
are made so sad by getting

210
00:09:40,190 --> 00:09:44,250
to 0 that it vastly
overcompensate the happiness

211
00:09:44,250 --> 00:09:47,900
you feel moving to $225 because
of the diminishing

212
00:09:47,900 --> 00:09:49,910
marginal utility.

213
00:09:49,910 --> 00:09:51,610
Because, basically, think
of it this way.

214
00:09:51,610 --> 00:09:53,290
Imagine it's your
actual income.

215
00:09:53,290 --> 00:09:55,450
Let's take the point about the
size of the gamble relative to

216
00:09:55,450 --> 00:09:56,340
income seriously.

217
00:09:56,340 --> 00:09:58,250
Imagine, literally, I was asking
you to gamble your

218
00:09:58,250 --> 00:09:59,820
entire income for the year.

219
00:09:59,820 --> 00:10:02,150
And if you lose, you
starve to death.

220
00:10:02,150 --> 00:10:04,070
And if you win, you get
to eat extra nice.

221
00:10:04,070 --> 00:10:08,360
Well, clearly, the disutility
of starving to death vastly

222
00:10:08,360 --> 00:10:11,590
outweighs the extra utility
to eating well.

223
00:10:11,590 --> 00:10:13,950
So, in that extreme example,
if this was your entire

224
00:10:13,950 --> 00:10:17,190
wealth, you can see why you
would have a situation where

225
00:10:17,190 --> 00:10:18,820
you wouldn't want to
take that gamble.

226
00:10:18,820 --> 00:10:21,760
Because if you lost,
you'd die.

227
00:10:21,760 --> 00:10:27,110
And, basically, risk aversion
arises because, basically,

228
00:10:27,110 --> 00:10:29,410
with diminishing marginal
utility you're

229
00:10:29,410 --> 00:10:30,730
made so much sadder.

230
00:10:30,730 --> 00:10:34,260
That steepness at the bottom,
you get so much sadder as you

231
00:10:34,260 --> 00:10:39,290
get towards 0 that it vastly
overcompensates the flatter

232
00:10:39,290 --> 00:10:43,030
part as you move above
your initial point.

233
00:10:43,030 --> 00:10:46,150
So, as you can see, you are
going to end up not wanting

234
00:10:46,150 --> 00:10:48,110
gambles even if they're fair.

235
00:10:48,110 --> 00:10:53,230
Gambles that are fair, that is
positive expected value, might

236
00:10:53,230 --> 00:10:57,860
still lead to a reduction in
your expected utility.

237
00:10:57,860 --> 00:10:59,760
Indeed, let me go further.

238
00:10:59,760 --> 00:11:03,310
You dislike this gamble so
much that if I said the

239
00:11:03,310 --> 00:11:06,840
following, I as your teacher am
going to force you to take

240
00:11:06,840 --> 00:11:09,740
this gamble-- imagine it's
like 100 years ago where

241
00:11:09,740 --> 00:11:11,830
teachers can beat students
and stuff--

242
00:11:11,830 --> 00:11:17,170
I'm going to force you take this
gamble unless you pay me,

243
00:11:17,170 --> 00:11:20,240
you would actually be
willing to pay me to

244
00:11:20,240 --> 00:11:23,310
avoid taking this gamble.

245
00:11:23,310 --> 00:11:24,560
How much would you pay me?

246
00:11:28,110 --> 00:11:30,890
Imagine utilities in dollar
terms. Imagine we're actually

247
00:11:30,890 --> 00:11:33,720
measuring utility in dollar
terms. How much would you pay

248
00:11:33,720 --> 00:11:37,160
me to avoid taking
this gamble.

249
00:11:37,160 --> 00:11:41,790
If I said you either take the
gamble, or you pay me.

250
00:11:41,790 --> 00:11:44,300
You're starting with
a utility of 100.

251
00:11:44,300 --> 00:11:44,800
Yeah?

252
00:11:44,800 --> 00:11:47,800
AUDIENCE: The difference between
the two utilities.

253
00:11:47,800 --> 00:11:48,550
PROFESSOR: Well, the difference

254
00:11:48,550 --> 00:11:49,530
between the two utilities.

255
00:11:49,530 --> 00:11:52,250
So utility is 100 here.

256
00:11:52,250 --> 00:11:56,740
Here utility is 7.5
squared, so 56.25.

257
00:11:56,740 --> 00:12:01,020
So you would actually
pay me $43.75 to

258
00:12:01,020 --> 00:12:02,090
avoid taking this gamble.

259
00:12:02,090 --> 00:12:03,120
Think about that.

260
00:12:03,120 --> 00:12:08,230
I've offered you a more than
fair bet, a very good bet,

261
00:12:08,230 --> 00:12:11,220
which, on average, will yield
you a positive $12.50.

262
00:12:11,220 --> 00:12:14,720
Yet you will pay me $43.75.

263
00:12:14,720 --> 00:12:18,390
You will pay almost half of your
entire wealth to avoid

264
00:12:18,390 --> 00:12:20,210
taking that gamble.

265
00:12:20,210 --> 00:12:22,300
That's pretty incredible
if you think about it.

266
00:12:22,300 --> 00:12:25,600
I've offered you a more than
fair bet, and yet you will pay

267
00:12:25,600 --> 00:12:28,390
me more than half your wealth,
almost half your wealth, to

268
00:12:28,390 --> 00:12:31,850
avoid taking that bet.

269
00:12:31,850 --> 00:12:32,960
So another way to see
this, let's look at

270
00:12:32,960 --> 00:12:33,930
this another way.

271
00:12:33,930 --> 00:12:37,800
How large would I have to make
the positive payoff for you to

272
00:12:37,800 --> 00:12:39,085
take the bet?

273
00:12:39,085 --> 00:12:41,120
Let's look at it that way.

274
00:12:41,120 --> 00:12:43,700
Right now I said you win
$125 with heads.

275
00:12:43,700 --> 00:12:46,853
How much would you have to win
with heads if you were going

276
00:12:46,853 --> 00:12:47,320
to take that bet?

277
00:12:47,320 --> 00:12:48,480
Yeah.

278
00:12:48,480 --> 00:12:50,267
And tell us how you
figured that out.

279
00:12:50,267 --> 00:12:53,000
AUDIENCE: Because you need to
have at least the same utility

280
00:12:53,000 --> 00:12:55,734
as you had before from the
unexpected utility.

281
00:12:55,734 --> 00:12:59,710
So more than half of his
per year utility

282
00:12:59,710 --> 00:13:01,718
would be 20 if he wins.

283
00:13:01,718 --> 00:13:03,510
20 squared is 400.

284
00:13:03,510 --> 00:13:04,860
[INAUDIBLE  PHRASE].

285
00:13:04,860 --> 00:13:05,090
PROFESSOR: Right.

286
00:13:05,090 --> 00:13:06,570
You'd need to win 300.

287
00:13:06,570 --> 00:13:10,170
Because I'd need to take your
utility to 20 if you win.

288
00:13:10,170 --> 00:13:13,150
Only then would you be willing
to take this gamble.

289
00:13:13,150 --> 00:13:16,760
So another way to say it is
that's how fair a gamble would

290
00:13:16,760 --> 00:13:18,436
need to be, how more than
fair it would need to be

291
00:13:18,436 --> 00:13:19,380
before you take it.

292
00:13:19,380 --> 00:13:25,100
You'd need me to pay off 3:1 on
a 50% chance before you'd

293
00:13:25,100 --> 00:13:27,080
take the bet.

294
00:13:27,080 --> 00:13:29,020
And this is just with a typical
looking utility

295
00:13:29,020 --> 00:13:31,510
function of the kind we worked
earlier in the semester.

296
00:13:31,510 --> 00:13:32,290
You didn't look at this earlier
in the semester and

297
00:13:32,290 --> 00:13:34,510
say, wow, that's a bizarre
utility function.

298
00:13:34,510 --> 00:13:37,080
We got sensible answers on our
problems, and problem sets,

299
00:13:37,080 --> 00:13:38,950
and tests, and things,
examples from

300
00:13:38,950 --> 00:13:39,520
square root of c.

301
00:13:39,520 --> 00:13:41,490
That seemed like a sensible
function.

302
00:13:41,490 --> 00:13:45,580
And yet it yields these
incredibly wild predictions

303
00:13:45,580 --> 00:13:48,220
that you would pay people almost
half of your wealth to

304
00:13:48,220 --> 00:13:50,330
avoid engaging in a more
than fair bet.

305
00:13:53,890 --> 00:13:57,920
And that you would need the odds
to be like 3:1 before you

306
00:13:57,920 --> 00:13:59,820
even consider taking a a bet.

307
00:13:59,820 --> 00:14:02,150
That's the power of uncertainty
and the power of

308
00:14:02,150 --> 00:14:03,140
risk aversion.

309
00:14:03,140 --> 00:14:06,020
Really, risk aversion, it's just
the power of diminishing

310
00:14:06,020 --> 00:14:08,410
marginal utility.

311
00:14:08,410 --> 00:14:11,120
The power of diminishing
marginal utility is so key to

312
00:14:11,120 --> 00:14:12,760
driving our decisions.

313
00:14:12,760 --> 00:14:15,870
It's the fact that that first
pizza means so much more to

314
00:14:15,870 --> 00:14:20,530
you than the fifth pizza, that
you really hate outcomes that

315
00:14:20,530 --> 00:14:22,680
don't let you get
the first pizza.

316
00:14:22,680 --> 00:14:27,050
And, as a result, you will pay
a lot to be forced into a

317
00:14:27,050 --> 00:14:29,970
situation where you don't
get any pizzas.

318
00:14:29,970 --> 00:14:32,580
You'll need to be paid a lot in
the state where you do win

319
00:14:32,580 --> 00:14:35,670
to deal with the state
where you don't.

320
00:14:35,670 --> 00:14:36,920
Questions about that?

321
00:14:39,310 --> 00:14:41,570
Now, we can change the example
in some interesting ways to

322
00:14:41,570 --> 00:14:42,200
understand it.

323
00:14:42,200 --> 00:14:46,920
So let's change the example to
say, instead, let's talk about

324
00:14:46,920 --> 00:14:51,510
some alternatives to this
example and how they affect

325
00:14:51,510 --> 00:14:52,330
our intuition.

326
00:14:52,330 --> 00:14:55,940
First alternative, imagine your
utility function instead

327
00:14:55,940 --> 00:15:00,040
of being square root of c, your
utility function was 0.1

328
00:15:00,040 --> 00:15:05,270
times c, a linear utility
function, not a non-linear

329
00:15:05,270 --> 00:15:06,520
utility function.

330
00:15:11,670 --> 00:15:16,360
We can now say that, in that
case, you actually would take

331
00:15:16,360 --> 00:15:17,610
the gamble.

332
00:15:20,780 --> 00:15:23,190
There's a 0.5% chance of 0.

333
00:15:26,340 --> 00:15:28,780
And I chose 0.1 times c, because
your initial utility

334
00:15:28,780 --> 00:15:31,020
is still 10 then.

335
00:15:31,020 --> 00:15:31,890
I normalized this.

336
00:15:31,890 --> 00:15:36,950
So starting with your bundle of
100 you still start at 10.

337
00:15:36,950 --> 00:15:38,080
It gives the same starting
point as

338
00:15:38,080 --> 00:15:40,560
the square root function.

339
00:15:40,560 --> 00:15:45,200
But now your expected utility
from his gamble is 0.5 times 0

340
00:15:45,200 --> 00:15:53,950
plus 0.5 times if you win 125,
your utility is 12.5.

341
00:15:53,950 --> 00:15:54,460
I'm sorry.

342
00:15:54,460 --> 00:15:59,020
It's 22.5.

343
00:15:59,020 --> 00:16:05,410
So your expected utility is
11.25 which is higher than

344
00:16:05,410 --> 00:16:07,530
your starting utility.

345
00:16:07,530 --> 00:16:10,210
So you would take this gamble.

346
00:16:10,210 --> 00:16:12,280
What's changed?

347
00:16:12,280 --> 00:16:13,660
AUDIENCE: No diminishing
marginal utility.

348
00:16:13,660 --> 00:16:15,880
PROFESSOR: No diminishing
marginal utility because now

349
00:16:15,880 --> 00:16:17,310
we are no longer risk averse.

350
00:16:17,310 --> 00:16:21,050
We are what we call
risk neutral.

351
00:16:21,050 --> 00:16:26,210
A linear utility function
yields risks neutrality.

352
00:16:26,210 --> 00:16:29,910
And once you're risk neutral,
you only care

353
00:16:29,910 --> 00:16:33,790
about expected value.

354
00:16:33,790 --> 00:16:36,700
Risk neutral consumers
would only care

355
00:16:36,700 --> 00:16:39,320
about expected value.

356
00:16:39,320 --> 00:16:41,650
And so a linear utility function
will lead to risk

357
00:16:41,650 --> 00:16:42,410
neutrality since you don't have

358
00:16:42,410 --> 00:16:43,450
diminishing marginal utility.

359
00:16:43,450 --> 00:16:44,605
Then you take any
bet that's fair.

360
00:16:44,605 --> 00:16:45,460
You don't care.

361
00:16:45,460 --> 00:16:47,820
You're indifferent between
winning a dollar and losing a

362
00:16:47,820 --> 00:16:48,960
dollar with this utility
function.

363
00:16:48,960 --> 00:16:53,200
It doesn't matter if
you go up or down.

364
00:16:53,200 --> 00:16:55,100
The joy you get from winning
is the same as the pain you

365
00:16:55,100 --> 00:16:56,170
get from losing.

366
00:16:56,170 --> 00:16:58,090
Whereas with this utility
function, the pain you get

367
00:16:58,090 --> 00:17:01,610
from losing exceeds the
joy from winning.

368
00:17:01,610 --> 00:17:07,180
We can see that graphically in
the next figure, Figure 20-2,

369
00:17:07,180 --> 00:17:09,500
the case of risk neutrality.

370
00:17:09,500 --> 00:17:16,359
Here, you start at point
A. You have 100, and

371
00:17:16,359 --> 00:17:19,900
your utility is 10.

372
00:17:19,900 --> 00:17:22,220
Now, I've offered you a gamble
where there's a 50% chance of

373
00:17:22,220 --> 00:17:27,170
getting 0 and a 50% chance of
getting B. Well, that yields

374
00:17:27,170 --> 00:17:32,030
an outcome of c, which
is a higher utility.

375
00:17:32,030 --> 00:17:36,290
So since your utility is linear,
you're risk neutral,

376
00:17:36,290 --> 00:17:37,830
and you'll take any fair bet.

377
00:17:40,830 --> 00:17:42,440
We can go further.

378
00:17:42,440 --> 00:17:50,090
What if utility, instead, was of
the form u equals c squared

379
00:17:50,090 --> 00:17:52,760
over 1,000?

380
00:17:52,760 --> 00:17:54,810
What if this was your
utility function?

381
00:17:54,810 --> 00:17:59,280
Once again, your initial
utility u of 100 is 10.

382
00:17:59,280 --> 00:18:02,850
It's the same starting point.

383
00:18:02,850 --> 00:18:07,440
But this is a utility function
which now if you do this

384
00:18:07,440 --> 00:18:17,560
gamble, your expected utility is
50% times 0 plus 50% times

385
00:18:17,560 --> 00:18:30,100
$225 squared over 1,000
which is 25.3.

386
00:18:30,100 --> 00:18:34,460
That's a huge increase in
utility from this gamble.

387
00:18:34,460 --> 00:18:39,880
So your expected utility with
the gamble is 25.3.

388
00:18:39,880 --> 00:18:43,480
It's a huge increase
in utility.

389
00:18:43,480 --> 00:18:47,830
And that's because this is an
individual where the shape of

390
00:18:47,830 --> 00:18:49,860
the utility function has change
where they don't have

391
00:18:49,860 --> 00:18:51,320
diminishing marginal
utility, they have

392
00:18:51,320 --> 00:18:53,770
increasing marginal utility.

393
00:18:53,770 --> 00:18:54,650
We've never worked with utility

394
00:18:54,650 --> 00:18:56,380
functions like this before.

395
00:18:56,380 --> 00:19:00,280
These are individuals
we call risk-loving.

396
00:19:00,280 --> 00:19:04,810
That is, they are made happier
by winning $1 than they are

397
00:19:04,810 --> 00:19:06,760
made sadder by losing $1.

398
00:19:06,760 --> 00:19:09,030
It's the opposite of all the
intuition we developed earlier

399
00:19:09,030 --> 00:19:09,970
in this course.

400
00:19:09,970 --> 00:19:12,550
It's a crazy utility function.

401
00:19:12,550 --> 00:19:14,960
But the notion of a risk-loving
utility function

402
00:19:14,960 --> 00:19:18,375
is one where literally $1 that
moves you up makes you happier

403
00:19:18,375 --> 00:19:20,630
than $1 that moves you down
makes you sadder.

404
00:19:20,630 --> 00:19:27,650
You can see that
in Figure 20-3.

405
00:19:27,650 --> 00:19:30,550
Here's a risk-loving
utility function.

406
00:19:30,550 --> 00:19:37,770
The individual starts at point
A. They have a choice of a

407
00:19:37,770 --> 00:19:44,400
gamble where they can have a 50%
chance of landing at 0 and

408
00:19:44,400 --> 00:19:45,330
a 50% chance--

409
00:19:45,330 --> 00:19:46,800
Jessica, that B should
be down at the

410
00:19:46,800 --> 00:19:48,550
intersection of dashed lines--

411
00:19:48,550 --> 00:19:51,305
a 50% chance of landing at B
at the intersection of the

412
00:19:51,305 --> 00:19:52,770
dashed lines.

413
00:19:52,770 --> 00:19:55,220
You take the average of
those two, and it's c.

414
00:19:55,220 --> 00:19:58,780
Their utility is way higher with
the gamble than it was

415
00:19:58,780 --> 00:20:00,870
without the gamble.

416
00:20:00,870 --> 00:20:01,560
In fact.

417
00:20:01,560 --> 00:20:03,540
we can go further.

418
00:20:03,540 --> 00:20:06,390
With a risk-loving person, they
would actually take an

419
00:20:06,390 --> 00:20:08,650
unfair bet.

420
00:20:08,650 --> 00:20:10,970
Consider the following bet.

421
00:20:10,970 --> 00:20:16,230
Tails you lose $100,
heads you win $75.

422
00:20:16,230 --> 00:20:18,455
That's a bet with a negative
expected value.

423
00:20:22,900 --> 00:20:25,520
Neither the risk averse nor the
risk neutral person would

424
00:20:25,520 --> 00:20:26,320
take that bet.

425
00:20:26,320 --> 00:20:27,490
But a risk-loving
person would.

426
00:20:27,490 --> 00:20:31,610
If you work out the math, that
bet gives them a gain in

427
00:20:31,610 --> 00:20:33,080
expected utility.

428
00:20:33,080 --> 00:20:35,380
That is a bet with a negative
expected value that gives them

429
00:20:35,380 --> 00:20:36,740
a gain in expected utility.

430
00:20:36,740 --> 00:20:37,845
Why is that?

431
00:20:37,845 --> 00:20:39,610
Because it's the opposite
of diminishing

432
00:20:39,610 --> 00:20:40,960
marginal utility intuition.

433
00:20:40,960 --> 00:20:45,740
They're made so much happier
by winning that they're

434
00:20:45,740 --> 00:20:46,910
willing to take a bet
even if it's a

435
00:20:46,910 --> 00:20:47,990
negative expected value.

436
00:20:47,990 --> 00:20:50,220
Just like the risk averse person
is made so much sadder

437
00:20:50,220 --> 00:20:52,310
by losing, they won't
take a bet even if

438
00:20:52,310 --> 00:20:54,000
it's more than fair.

439
00:20:54,000 --> 00:20:57,650
So you can actually develop all
the opposite predictions

440
00:20:57,650 --> 00:20:58,880
from a risk-loving person.

441
00:20:58,880 --> 00:21:02,150
They'll even take
unfair gambles.

442
00:21:02,150 --> 00:21:03,830
Now, by the way, I skipped over
your earlier question

443
00:21:03,830 --> 00:21:04,910
about risk neutrality.

444
00:21:04,910 --> 00:21:07,700
With risk neutrality, you see it
doesn't matter if you do it

445
00:21:07,700 --> 00:21:09,920
100 times or one time.

446
00:21:09,920 --> 00:21:13,860
If you're risk neutral, you
should take the bet anytime,

447
00:21:13,860 --> 00:21:16,530
because the expected value
is still positive.

448
00:21:16,530 --> 00:21:18,145
Now, you're thinking about
risk aversion where, in

449
00:21:18,145 --> 00:21:21,100
substance, you're more
confident as

450
00:21:21,100 --> 00:21:22,150
the numbers go up.

451
00:21:22,150 --> 00:21:24,380
But if you're risk neutral,
you'll take it no matter how

452
00:21:24,380 --> 00:21:27,090
many times I offer
you that bet.

453
00:21:27,090 --> 00:21:32,870
So to extend this further, let's
go to a third extension

454
00:21:32,870 --> 00:21:35,460
which will develop this
intuition further.

455
00:21:35,460 --> 00:21:39,000
Now imagine that I offer you
guys a different gamble.

456
00:21:39,000 --> 00:21:40,960
And, once again, I really want
you to answer honestly.

457
00:21:40,960 --> 00:21:41,925
Don't try to game me.

458
00:21:41,925 --> 00:21:43,120
Answer honestly.

459
00:21:43,120 --> 00:21:54,380
Now the gamble is if I flip a
coin, tails you lose $1, heads

460
00:21:54,380 --> 00:21:57,510
you win $1.25.

461
00:21:57,510 --> 00:22:00,470
Now how many of you would
take that gamble.

462
00:22:00,470 --> 00:22:02,770
How many would not
take that gamble?

463
00:22:02,770 --> 00:22:03,750
OK.

464
00:22:03,750 --> 00:22:04,940
I hope you're answering
honestly.

465
00:22:04,940 --> 00:22:07,870
But maybe you're just thinking
ahead and realizing that that

466
00:22:07,870 --> 00:22:08,680
gamble is very different.

467
00:22:08,680 --> 00:22:11,350
And why are people more willing
to take that gamble

468
00:22:11,350 --> 00:22:13,710
than they were willing to take
the previous gamble, the same

469
00:22:13,710 --> 00:22:14,390
risk averse people.

470
00:22:14,390 --> 00:22:15,156
Yeah.

471
00:22:15,156 --> 00:22:19,440
AUDIENCE: The difference
in [INAUDIBLE PHRASE].

472
00:22:19,440 --> 00:22:19,970
PROFESSOR: Exactly.

473
00:22:19,970 --> 00:22:23,535
In particular, the utility
function is locally linear.

474
00:22:28,350 --> 00:22:31,400
Let's go back to Figure 20-1.

475
00:22:31,400 --> 00:22:34,760
As you get closer and closer
to A, you could draw,

476
00:22:34,760 --> 00:22:37,300
essentially, a linear segment.

477
00:22:37,300 --> 00:22:42,810
So for an infinitesimal bet,
utility is linear.

478
00:22:42,810 --> 00:22:45,080
So it's linear at point A.

479
00:22:45,080 --> 00:22:48,740
So for small bets, you
become risk neutral.

480
00:22:48,740 --> 00:22:52,600
Even a risk averse person moves
towards risk neutrality

481
00:22:52,600 --> 00:22:55,750
as the bet is small relative
to their resources.

482
00:22:55,750 --> 00:22:57,380
This was the point that
you were making.

483
00:22:57,380 --> 00:22:59,140
Basically if you're a rich
person, you'd probably be

484
00:22:59,140 --> 00:23:01,280
happy to take the $100
and $125 thing.

485
00:23:01,280 --> 00:23:02,280
I'd be happy to do that.

486
00:23:02,280 --> 00:23:04,310
I'm a rich guy.

487
00:23:04,310 --> 00:23:05,890
I'd be happy to do that.

488
00:23:05,890 --> 00:23:10,550
So, basically, what determines
your willingness to take a bet

489
00:23:10,550 --> 00:23:12,880
is going to be about
what's at stake

490
00:23:12,880 --> 00:23:14,130
relative to your resources.

491
00:23:17,660 --> 00:23:21,500
And what you can see is that
if you solve the math here,

492
00:23:21,500 --> 00:23:24,130
that basically expected utility
even with a square

493
00:23:24,130 --> 00:23:30,890
root of c is positive for
that smaller gamble.

494
00:23:30,890 --> 00:23:33,830
Because as it gets smaller
relative to the $100 you start

495
00:23:33,830 --> 00:23:36,460
with, you become roughly
risk neutral.

496
00:23:36,460 --> 00:23:39,250
And then you'll go ahead
and take the gamble.

497
00:23:39,250 --> 00:23:42,750
So at the end of the day what's
going to determine

498
00:23:42,750 --> 00:23:47,030
whether you're going to take a
gamble is going to be your

499
00:23:47,030 --> 00:23:53,090
level of risk aversion and the
size of the risk you're taking

500
00:23:53,090 --> 00:23:55,140
relative to your resources.

501
00:23:55,140 --> 00:23:59,110
The more risk averse you are,
and the bigger the gamble, the

502
00:23:59,110 --> 00:24:01,315
less likely you are to take it
at a given level of fairness.

503
00:24:04,530 --> 00:24:05,780
Questions about that?

504
00:24:08,490 --> 00:24:09,210
All right.

505
00:24:09,210 --> 00:24:12,640
So now that we all understand
expected utility theory.

506
00:24:12,640 --> 00:24:14,670
Now we're going to go on and
talk about why this matters in

507
00:24:14,670 --> 00:24:16,340
the real world and
how we use it.

508
00:24:16,340 --> 00:24:17,940
And I want to talk, in
particular, about two

509
00:24:17,940 --> 00:24:22,880
applications, insurance
and the lottery.

510
00:24:22,880 --> 00:24:25,630
Let's start by talking
about insurance and

511
00:24:25,630 --> 00:24:27,640
why people have insurance.

512
00:24:27,640 --> 00:24:31,150
Because, in fact, given what
we learned in this lecture,

513
00:24:31,150 --> 00:24:33,250
there would be no reason
for insurance.

514
00:24:33,250 --> 00:24:35,336
This lecture tells us why
people have insurance.

515
00:24:37,910 --> 00:24:40,510
Because there's diminishing
marginal utility, and you're

516
00:24:40,510 --> 00:24:43,100
made so much sadder with a
negative outcome, you're

517
00:24:43,100 --> 00:24:45,560
willing to pay you avoid it.

518
00:24:45,560 --> 00:24:47,280
Remember we talked about that
you would be willing to pay

519
00:24:47,280 --> 00:24:50,400
almost $44 to avoid being
forced to take that bet?

520
00:24:50,400 --> 00:24:52,580
That's what insurance does.

521
00:24:52,580 --> 00:24:56,230
Insurance allows you to
avoid taking gambles.

522
00:24:56,230 --> 00:24:57,390
That's what you can think
of insurance as.

523
00:24:57,390 --> 00:24:59,300
It's a way to avoid
taking a gamble.

524
00:24:59,300 --> 00:25:01,130
You're gambling you're
going to get sick.

525
00:25:01,130 --> 00:25:03,670
You're gambling your house
is going to burn down.

526
00:25:03,670 --> 00:25:05,060
These are gambles you
face that are

527
00:25:05,060 --> 00:25:07,590
forced on you by nature.

528
00:25:07,590 --> 00:25:09,500
What insurance does is
allow you to avoid

529
00:25:09,500 --> 00:25:10,530
taking those gambles.

530
00:25:10,530 --> 00:25:13,960
And just like you'd pay me to
avoid the $100, $125 gamble,

531
00:25:13,960 --> 00:25:16,940
you're paying Aetna to avoid
gambling that you might have

532
00:25:16,940 --> 00:25:19,190
to go to the hospital.

533
00:25:19,190 --> 00:25:21,990
So let's say there's a
25-year-old who is deciding

534
00:25:21,990 --> 00:25:23,470
whether to buy health
insurance.

535
00:25:23,470 --> 00:25:24,760
And let's say they're
25-year-old

536
00:25:24,760 --> 00:25:26,320
guy, totally healthy.

537
00:25:26,320 --> 00:25:27,490
I say guy because there's
no risk they're

538
00:25:27,490 --> 00:25:28,570
going to have a kid.

539
00:25:28,570 --> 00:25:32,130
So he's basically totally
healthy, basically zero chance

540
00:25:32,130 --> 00:25:33,422
they're going to use the
doctor except if they

541
00:25:33,422 --> 00:25:35,470
get hit by a car.

542
00:25:35,470 --> 00:25:38,420
So imagine the situation is that
you've got 25-year-old

543
00:25:38,420 --> 00:25:45,440
with an income of $40,000.

544
00:25:45,440 --> 00:25:48,320
And let's say that there's
a 1% chance that they'll

545
00:25:48,320 --> 00:25:48,960
get hit by a car.

546
00:25:48,960 --> 00:25:50,340
It is Cambridge after all.

547
00:25:50,340 --> 00:25:52,190
So every time you cross the
street, there's a 1% chance

548
00:25:52,190 --> 00:25:53,440
you get hit by a car.

549
00:25:57,380 --> 00:26:00,440
And if you get hit by a car,
you're going to suffer $30,000

550
00:26:00,440 --> 00:26:01,690
in hospital bills.

551
00:26:09,480 --> 00:26:13,060
And let's say your utility
function is square root of c.

552
00:26:13,060 --> 00:26:14,310
So you're a risk averse guy.

553
00:26:17,270 --> 00:26:24,260
So let's say that I then come
to you and say, look, each

554
00:26:24,260 --> 00:26:30,320
year there's an expected
cost to you of getting

555
00:26:30,320 --> 00:26:32,250
hit by car of $300.

556
00:26:32,250 --> 00:26:33,160
How did I calculate that?

557
00:26:33,160 --> 00:26:36,280
Well, every year there's a
1% chance you get hit.

558
00:26:36,280 --> 00:26:38,020
They're independent
draws, let's say.

559
00:26:38,020 --> 00:26:39,600
If you get hit this year,
it doesn't mean

560
00:26:39,600 --> 00:26:41,000
suddenly you're safer.

561
00:26:41,000 --> 00:26:41,710
It's random.

562
00:26:41,710 --> 00:26:43,140
It's just crazy drivers.

563
00:26:43,140 --> 00:26:45,630
So there's a 1% chance you're
going to get hit every year.

564
00:26:45,630 --> 00:26:48,133
And if you get hit, there's a
$30,000 cost. So every year

565
00:26:48,133 --> 00:26:49,880
there's an expected
cost to you--

566
00:26:49,880 --> 00:26:51,870
the opposite of expected value
is expected cost--

567
00:26:51,870 --> 00:26:53,590
of $300.

568
00:26:53,590 --> 00:26:55,910
So let's say I offered to sell
you insurance for $300.

569
00:26:55,910 --> 00:26:59,740
I offered to sell you insurance
in a way where, on

570
00:26:59,740 --> 00:27:04,890
average, if you lived an
infinite number of years, you

571
00:27:04,890 --> 00:27:08,020
would pay out in premiums what
you'd get in benefits.

572
00:27:08,020 --> 00:27:11,080
If you paid $300 a year and
lived forever or lived for

573
00:27:11,080 --> 00:27:14,520
many, many years-- the law of
large numbers enough years--

574
00:27:14,520 --> 00:27:16,460
then basically you would pay out
in premiums what you would

575
00:27:16,460 --> 00:27:18,850
collect in benefits.

576
00:27:18,850 --> 00:27:20,830
You'd get hit once
every 100 years.

577
00:27:20,830 --> 00:27:23,850
And ever 100 years you would
have paid $30,000 in premiums,

578
00:27:23,850 --> 00:27:26,330
and you'd collect $30,000
in benefits.

579
00:27:26,330 --> 00:27:30,880
So that's what we call
actuarially fair insurance.

580
00:27:30,880 --> 00:27:43,030
Actuarially fair insurance is
insurance where the price of

581
00:27:43,030 --> 00:27:50,560
the insurance equals the
probability of the bad outcome

582
00:27:50,560 --> 00:27:54,770
times the cost of
the bad outcome.

583
00:27:57,750 --> 00:28:00,140
That's actuarially fair
insurance where the price you

584
00:28:00,140 --> 00:28:02,005
pay is the probability of the
bad outcome times the cost of

585
00:28:02,005 --> 00:28:02,550
the bad outcome.

586
00:28:02,550 --> 00:28:05,540
That's fair because, over a
large enough population, the

587
00:28:05,540 --> 00:28:10,250
premiums that get paid in will
get paid out in the form of

588
00:28:10,250 --> 00:28:13,120
claims.

589
00:28:13,120 --> 00:28:18,460
Now, let's ask what is your
utility if you do not or do

590
00:28:18,460 --> 00:28:19,190
buy insurance.

591
00:28:19,190 --> 00:28:20,780
So for the first thing you
say if I'm a 25-year-old.

592
00:28:20,780 --> 00:28:21,240
Screw it.

593
00:28:21,240 --> 00:28:22,170
I'm never going to
get hit by a car.

594
00:28:22,170 --> 00:28:23,430
I'm not going to
buy insurance.

595
00:28:23,430 --> 00:28:26,680
What's your utility
with no insurance?

596
00:28:26,680 --> 00:28:32,410
Well, if you have no insurance,
there's a 1%

597
00:28:32,410 --> 00:28:36,020
chance, 0.01, that you'll
lose $30,000.

598
00:28:36,020 --> 00:28:37,305
You'll get hit by a car
and lose $30,000.

599
00:28:37,305 --> 00:28:39,710
Your income is $40,000.

600
00:28:39,710 --> 00:28:42,890
So there's a 1% chance that
you'll end up with a utility,

601
00:28:42,890 --> 00:28:47,910
which is the square
root of 10,000.

602
00:28:47,910 --> 00:28:55,850
And there's a 99% chance you'll
end up with a utility

603
00:28:55,850 --> 00:29:00,120
that's the square
root of 40,000.

604
00:29:00,120 --> 00:29:09,200
You work this out, and the
answer is you get 199.

605
00:29:09,200 --> 00:29:14,700
Utility without insurance is 199
which is pretty close to

606
00:29:14,700 --> 00:29:16,510
utility just if you weren't
going to get hit by the car.

607
00:29:16,510 --> 00:29:18,660
Because it's so rare that
you get hit by the car.

608
00:29:18,660 --> 00:29:23,160
So utility is 199 without
insurance.

609
00:29:23,160 --> 00:29:30,170
Now, let's ask the question, how
much would you be willing

610
00:29:30,170 --> 00:29:33,520
to pay to have insurance?

611
00:29:33,520 --> 00:29:34,980
How do we figure that out?

612
00:29:34,980 --> 00:29:37,560
$300 is the actuarially
fair premium.

613
00:29:37,560 --> 00:29:38,500
But now let's do a different
question.

614
00:29:38,500 --> 00:29:40,560
I'm an insurance company, and
I want to make money.

615
00:29:40,560 --> 00:29:42,160
I don't want to just charge the
actuarially fair premium.

616
00:29:42,160 --> 00:29:43,770
The insurance company
makes no money with

617
00:29:43,770 --> 00:29:45,400
this premium of $300.

618
00:29:45,400 --> 00:29:47,870
So the insurance company
wants to make money.

619
00:29:47,870 --> 00:29:50,010
How would we figure out how much
would you be willing to

620
00:29:50,010 --> 00:29:52,340
pay, this 25-year-old,
be willing

621
00:29:52,340 --> 00:29:54,800
to pay to get insurance?

622
00:29:54,800 --> 00:29:57,590
How do we figure that out?

623
00:29:57,590 --> 00:29:58,950
Yeah.

624
00:29:58,950 --> 00:30:02,020
AUDIENCE: Maybe you could keep
the utility function constant.

625
00:30:02,020 --> 00:30:03,315
PROFESSOR: Keep the utility
value constant.

626
00:30:03,315 --> 00:30:04,140
AUDIENCE: Value, yes.

627
00:30:04,140 --> 00:30:04,730
PROFESSOR: Exactly.

628
00:30:04,730 --> 00:30:08,520
You'd have ask well, how much
would I be willing to pay to

629
00:30:08,520 --> 00:30:10,420
have insurance which would
protect me and leave me at the

630
00:30:10,420 --> 00:30:12,190
same utility level.

631
00:30:12,190 --> 00:30:13,320
Obviously it would have to
be a little bit higher.

632
00:30:13,320 --> 00:30:15,770
But let's just set it equal.

633
00:30:15,770 --> 00:30:19,770
So, in other words, if I bought
insurance, my utility

634
00:30:19,770 --> 00:30:28,200
with insurance, there's a
1% chance that I will

635
00:30:28,200 --> 00:30:29,490
get hit by the car.

636
00:30:29,490 --> 00:30:30,980
In that case, what
happens to me?

637
00:30:30,980 --> 00:30:36,290
Well, if I get hit by the
car, I get $10,000.

638
00:30:36,290 --> 00:30:37,620
I make $40,000.

639
00:30:37,620 --> 00:30:39,360
I lose $30,000.

640
00:30:39,360 --> 00:30:40,740
Let me actually write it out.

641
00:30:40,740 --> 00:30:42,900
If I get hit by the car,
what happens to me?

642
00:30:42,900 --> 00:30:46,580
Well, I make $40,000.

643
00:30:46,580 --> 00:30:48,070
I always make $40,000
each year.

644
00:30:52,640 --> 00:30:55,970
I lose $30,000, because
I get hit by the car.

645
00:30:55,970 --> 00:30:58,760
But then the insurance company
pays me $30,000.

646
00:30:58,760 --> 00:31:00,230
They pay off my debts.

647
00:31:00,230 --> 00:31:03,400
So then I gain $30,000.

648
00:31:03,400 --> 00:31:05,250
So these things cancel.

649
00:31:05,250 --> 00:31:07,520
But I have to pay the insurance
company premium.

650
00:31:07,520 --> 00:31:09,310
So I have to pay
some amount x.

651
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

652
00:31:20,200 --> 00:31:23,180
still have to pay the insurance
company premium.

653
00:31:23,180 --> 00:31:24,640
I have to pay them whether
I get hit or not.

654
00:31:24,640 --> 00:31:25,200
It's insurance.

655
00:31:25,200 --> 00:31:28,050
I pay them either way.

656
00:31:28,050 --> 00:31:29,800
So that's my utility.

657
00:31:29,800 --> 00:31:35,790
So my expected utility
with insurance is the

658
00:31:35,790 --> 00:31:37,040
sum of these two.

659
00:31:39,270 --> 00:31:42,730
And I want to set that
equal to 199.

660
00:31:42,730 --> 00:31:46,380
I want to say what x am I
willing to pay that would

661
00:31:46,380 --> 00:31:50,590
leave me at the same utility as
if I was uninsured as per

662
00:31:50,590 --> 00:31:52,930
the answer here?

663
00:31:52,930 --> 00:31:57,080
Well, it turns out that if you
do, that if you solve this,

664
00:31:57,080 --> 00:32:05,040
you get that x equals 399.

665
00:32:05,040 --> 00:32:10,870
That is you would pay $399
for insurance that has a

666
00:32:10,870 --> 00:32:14,350
value of only $300.

667
00:32:14,350 --> 00:32:18,220
You'd pay $399 for insurance
even though the actuarially

668
00:32:18,220 --> 00:32:20,590
fair price is $300.

669
00:32:20,590 --> 00:32:22,870
You would pay you insurance
company $99 more than they

670
00:32:22,870 --> 00:32:25,400
expect to pay out to you.

671
00:32:25,400 --> 00:32:25,910
Why?

672
00:32:25,910 --> 00:32:27,610
Because you're risk averse.

673
00:32:27,610 --> 00:32:31,220
Because you're made so much
sadder than being left with

674
00:32:31,220 --> 00:32:35,200
$10,000 than you are by
having to pay $300.

675
00:32:35,200 --> 00:32:36,960
If it doesn't work out,
you pay $300.

676
00:32:36,960 --> 00:32:37,370
Who cares?

677
00:32:37,370 --> 00:32:39,050
That's tiny compared
to your income.

678
00:32:39,050 --> 00:32:41,200
But if it does work
out, you're safe

679
00:32:41,200 --> 00:32:44,200
from having to starve.

680
00:32:44,200 --> 00:32:45,520
You pay $400, I'm sorry.

681
00:32:45,520 --> 00:32:47,440
You pay $399.

682
00:32:47,440 --> 00:32:50,380
You're like, look, I'll be
bummed if I have to pay $400.

683
00:32:50,380 --> 00:32:52,400
That's a percent of my
income basically.

684
00:32:52,400 --> 00:32:53,990
That would be a shame to pay
a percent of my income for

685
00:32:53,990 --> 00:32:55,210
something that doesn't happen.

686
00:32:55,210 --> 00:32:57,890
But, boy, would I be happy in
that 1 in 100 chance where I

687
00:32:57,890 --> 00:33:01,530
get hit by a car when
I'm not out $30,000.

688
00:33:01,530 --> 00:33:04,690
So you will pay $399
for insurance

689
00:33:04,690 --> 00:33:07,120
that's only worth $300.

690
00:33:07,120 --> 00:33:10,735
That extra $99 we call
a risk premium.

691
00:33:19,620 --> 00:33:22,230
We call that a risk premium.

692
00:33:22,230 --> 00:33:24,780
The extra $99, we call
a risk premium.

693
00:33:24,780 --> 00:33:30,470
That is the amount that you are
willing to pay above and

694
00:33:30,470 --> 00:33:34,260
beyond the fair price, because
you're risk averse.

695
00:33:34,260 --> 00:33:38,780
And what you should go home and
show yourself using the

696
00:33:38,780 --> 00:33:42,680
same kind of mathematics is
that, for example, the risk

697
00:33:42,680 --> 00:33:47,440
premium will rise the
bigger the loss is.

698
00:33:47,440 --> 00:33:48,470
Hopefully you can see the
intuition on that.

699
00:33:48,470 --> 00:33:51,250
The bigger the loss is for a
given level of income the

700
00:33:51,250 --> 00:33:52,200
bigger the risk is.

701
00:33:52,200 --> 00:33:54,240
Likewise, for a given
loss, the risk

702
00:33:54,240 --> 00:33:56,840
premium falls with income.

703
00:33:56,840 --> 00:33:59,430
So the bigger is the loss of
relative to income the more

704
00:33:59,430 --> 00:34:00,680
risk premium you're
willing to pay.

705
00:34:03,300 --> 00:34:06,470
You should also, obviously, see
that the more risk averse

706
00:34:06,470 --> 00:34:08,830
you are, the bigger premium
you're willing to pay.

707
00:34:08,830 --> 00:34:13,560
A risk neutral person would
not pay a risk premium.

708
00:34:13,560 --> 00:34:15,760
Only a risk averse
person will.

709
00:34:15,760 --> 00:34:17,719
So the more risk averse you are,
the bigger risk premium

710
00:34:17,719 --> 00:34:20,000
you'll pay, and the
bigger the loss is

711
00:34:20,000 --> 00:34:20,810
relative to your income.

712
00:34:20,810 --> 00:34:23,080
These are the same principles
we talked about before.

713
00:34:26,010 --> 00:34:29,120
So the $43.75 we were willing to
pay to avoid that gamble I

714
00:34:29,120 --> 00:34:33,050
was going to force on you, that
was the risk premium.

715
00:34:33,050 --> 00:34:36,300
You were willing to pay $44
to avoid that gamble.

716
00:34:36,300 --> 00:34:40,179
Here, you're willing to pay
$99 to avoid the risk of

717
00:34:40,179 --> 00:34:44,389
ending up in that bad state
where you get hit by the car.

718
00:34:44,389 --> 00:34:46,040
And that's why people
buy insurance.

719
00:34:46,040 --> 00:34:49,239
And that's why insurance
companies make

720
00:34:49,239 --> 00:34:51,130
ungodly amounts of money.

721
00:34:51,130 --> 00:34:53,250
In the US we have a health
insurance industry, for

722
00:34:53,250 --> 00:34:58,620
example, that earns about
$800 billion a year.

723
00:34:58,620 --> 00:34:59,830
Why do they make
all that money?

724
00:34:59,830 --> 00:35:01,600
Because people are risk averse,
and they're willing to

725
00:35:01,600 --> 00:35:04,130
pay to have someone else
bear the risk of

726
00:35:04,130 --> 00:35:07,370
their injury or illness.

727
00:35:07,370 --> 00:35:09,770
Any questions about that?

728
00:35:09,770 --> 00:35:12,180
Now, I don't mean by that to
say, insurance is a bad thing,

729
00:35:12,180 --> 00:35:13,360
and we shouldn't do it.

730
00:35:13,360 --> 00:35:15,310
Risk aversion is the nature
of our utility functions.

731
00:35:15,310 --> 00:35:17,260
We should be willing to
pay a risk premium.

732
00:35:17,260 --> 00:35:19,170
It's just that you need to
understand why, in fact, it

733
00:35:19,170 --> 00:35:23,030
makes sense to have insurance
in that case.

734
00:35:23,030 --> 00:35:25,200
The second application
is the lottery.

735
00:35:32,300 --> 00:35:35,460
The lottery is a total ripoff.

736
00:35:35,460 --> 00:35:37,005
I hope you knew this already.

737
00:35:37,005 --> 00:35:39,580
The expected value
of $1 lottery

738
00:35:39,580 --> 00:35:42,230
ticket is roughly $0.50.

739
00:35:42,230 --> 00:35:44,860
So for every $1 you spend in the
lottery, in expectation,

740
00:35:44,860 --> 00:35:46,870
you get about $0.50 back.

741
00:35:46,870 --> 00:35:52,930
This is an incredibly bad bet,
incredibly unfair, an

742
00:35:52,930 --> 00:35:55,160
incredibly unfair bet.

743
00:35:55,160 --> 00:35:59,930
On average, you lose $0.50
for every $1 you bet.

744
00:35:59,930 --> 00:36:04,700
So, basically, despite that,
lotteries are wildly popular.

745
00:36:04,700 --> 00:36:06,790
They've become a huge source
of revenue for state

746
00:36:06,790 --> 00:36:07,820
governments.

747
00:36:07,820 --> 00:36:09,900
A lot of the money that state
governments now take in is

748
00:36:09,900 --> 00:36:12,120
through state lotteries.

749
00:36:12,120 --> 00:36:16,860
What accounts for the fact that
lotteries are so popular?

750
00:36:16,860 --> 00:36:19,900
Well, there's four different
theories for why lotteries are

751
00:36:19,900 --> 00:36:21,510
so popular.

752
00:36:21,510 --> 00:36:25,030
The first is that people
are risk-loving.

753
00:36:25,030 --> 00:36:27,120
We have it all wrong.

754
00:36:27,120 --> 00:36:28,560
Actually people like taking
risks, and the

755
00:36:28,560 --> 00:36:29,810
lottery feeds that.

756
00:36:32,300 --> 00:36:34,110
This, of course, we can
immediately rule out.

757
00:36:34,110 --> 00:36:34,990
How?

758
00:36:34,990 --> 00:36:37,920
How do we know this is wrong?

759
00:36:37,920 --> 00:36:39,200
That the answer is that people
play the lottery because

760
00:36:39,200 --> 00:36:40,195
they're risk-loving.

761
00:36:40,195 --> 00:36:42,512
How do we know people
aren't risk-loving?

762
00:36:42,512 --> 00:36:44,380
AUDIENCE: The same people don't
take [UNINTELLIGIBLE]

763
00:36:44,380 --> 00:36:45,957
PROFESSOR: And they spend
$800 billion a

764
00:36:45,957 --> 00:36:47,930
year on health insurance.

765
00:36:47,930 --> 00:36:50,947
Basically, as a society, we
spend, in total, about $1.5

766
00:36:50,947 --> 00:36:54,000
trillion a year on insuring
various risks that face us.

767
00:36:54,000 --> 00:36:56,080
We're not risk-loving.

768
00:36:56,080 --> 00:36:59,620
So that's clearly
not the answer.

769
00:36:59,620 --> 00:37:01,840
However, there's
an alternative.

770
00:37:01,840 --> 00:37:09,280
People could basically alternate
between risk-loving

771
00:37:09,280 --> 00:37:10,530
and risk-aversion.

772
00:37:12,460 --> 00:37:15,230
This is a theory due to Milton
Friedman, the famous economist

773
00:37:15,230 --> 00:37:17,680
from Chicago and a co-author
named Savage, the

774
00:37:17,680 --> 00:37:20,960
Friedman-Savage preferences,
where the notion is that

775
00:37:20,960 --> 00:37:27,030
basically people are risk averse
over small gambles but

776
00:37:27,030 --> 00:37:29,060
risk-loving over
large gambles.

777
00:37:29,060 --> 00:37:31,980
So to see that, go last
figure in the graph.

778
00:37:31,980 --> 00:37:34,580
This is sort of a complicated
case.

779
00:37:34,580 --> 00:37:39,000
Basically, the notion is if you
take someone, they have a

780
00:37:39,000 --> 00:37:42,280
utility function which is
initially risk averse and then

781
00:37:42,280 --> 00:37:43,470
becomes risk-loving.

782
00:37:43,470 --> 00:37:48,040
That is in the segment between
W1 and W3, that looks like a

783
00:37:48,040 --> 00:37:50,140
risk averse utility function.

784
00:37:50,140 --> 00:37:53,060
But once you get above W3, it
looks like a risk-loving

785
00:37:53,060 --> 00:37:54,940
utility function.

786
00:37:54,940 --> 00:38:01,550
So the notion is that for things
which can make me very

787
00:38:01,550 --> 00:38:04,060
poor, I'm risk averse.

788
00:38:04,060 --> 00:38:08,230
I want to insure against events
which will leave me in

789
00:38:08,230 --> 00:38:09,380
that bottom segment.

790
00:38:09,380 --> 00:38:13,910
But once I'm going to be
above W3, then great.

791
00:38:13,910 --> 00:38:14,880
I'm happy to take risks.

792
00:38:14,880 --> 00:38:16,130
Then I become risk-loving.

793
00:38:18,160 --> 00:38:24,900
Now, this is a not crazy idea.

794
00:38:24,900 --> 00:38:28,380
Graphically, what I'm showing
you here, is that b* is

795
00:38:28,380 --> 00:38:30,610
utility without the gamble
and b is with.

796
00:38:30,610 --> 00:38:32,740
So you see you're happier
without the gamble when your

797
00:38:32,740 --> 00:38:33,820
income is low.

798
00:38:33,820 --> 00:38:36,380
Once your income is a lot
higher, you're happier with

799
00:38:36,380 --> 00:38:40,510
the gamble at d then you are
without the gamble at d*.

800
00:38:40,510 --> 00:38:42,550
That's not a crazy theory.

801
00:38:42,550 --> 00:38:44,860
The notion is that once I'm
rich enough, I become

802
00:38:44,860 --> 00:38:46,110
risk-loving.

803
00:38:46,110 --> 00:38:49,840
But when I'm poor, I don't
want to take the risks.

804
00:38:49,840 --> 00:38:52,750
The problem is that this is
inconsistent with lottery

805
00:38:52,750 --> 00:38:54,400
behavior in the following
sense.

806
00:38:54,400 --> 00:38:57,200
Most people who play
the lottery don't

807
00:38:57,200 --> 00:38:58,740
play the Mega Millions.

808
00:38:58,740 --> 00:39:01,035
They play tiny scratch
lotteries where you

809
00:39:01,035 --> 00:39:04,730
bet $1 to win $10.

810
00:39:04,730 --> 00:39:07,580
And people spend huge amounts
of money on lotteries with

811
00:39:07,580 --> 00:39:10,000
very, very low payoffs.

812
00:39:10,000 --> 00:39:12,950
That is inconsistent
with this.

813
00:39:12,950 --> 00:39:14,826
Because this would say that
you'd only play lotteries that

814
00:39:14,826 --> 00:39:15,970
have big payoffs.

815
00:39:15,970 --> 00:39:18,520
Lotteries that have small
payoffs, once again, there's

816
00:39:18,520 --> 00:39:21,460
no reason to play that and
still buy insurance.

817
00:39:21,460 --> 00:39:24,550
So if you're buying insurance
against being low income, why

818
00:39:24,550 --> 00:39:26,190
are you playing these small
lotteries that are a ripoff.

819
00:39:26,190 --> 00:39:28,290
Because those small ones
are a ripoff too.

820
00:39:28,290 --> 00:39:30,340
So the existence of the fact
that the most popular

821
00:39:30,340 --> 00:39:32,350
lotteries are actually the
small lotteries is

822
00:39:32,350 --> 00:39:34,200
inconsistent with this
explanation.

823
00:39:34,200 --> 00:39:35,306
Yeah.

824
00:39:35,306 --> 00:39:36,680
AUDIENCE: So I'm confused.

825
00:39:36,680 --> 00:39:39,430
Is it risk-loving on
large gambles?

826
00:39:39,430 --> 00:39:40,695
PROFESSOR: Yeah, risk-loving
on large gambles.

827
00:39:44,860 --> 00:39:46,390
It's not the size
of the gamble.

828
00:39:46,390 --> 00:39:48,760
You're risk-loving on gambles
which leave you in a high

829
00:39:48,760 --> 00:39:50,890
wealth state.

830
00:39:50,890 --> 00:39:53,220
The point is that if
I'm gambling over

831
00:39:53,220 --> 00:39:55,035
winning Mega Millions.

832
00:39:55,035 --> 00:39:56,320
Yeah, I'm a little
risk averse.

833
00:39:56,320 --> 00:39:59,000
But the truth is winning Mega
Millions would make me so

834
00:39:59,000 --> 00:40:00,770
happy that I could move into
the risk-loving part of my

835
00:40:00,770 --> 00:40:02,310
utility function.

836
00:40:02,310 --> 00:40:04,940
But this would not explain why
people ever play something

837
00:40:04,940 --> 00:40:07,470
that pays off $100.

838
00:40:07,470 --> 00:40:10,190
This is a fancy way of the
intuition you probably have.

839
00:40:10,190 --> 00:40:10,900
It's I'd think differently about
something which would

840
00:40:10,900 --> 00:40:13,010
completely change my life and
make me a multi-billionaire,

841
00:40:13,010 --> 00:40:15,730
that's something that would make
me raise me, than the bet

842
00:40:15,730 --> 00:40:17,560
I offered you guys before.

843
00:40:17,560 --> 00:40:19,590
People are systematically taking
terrible bets like the

844
00:40:19,590 --> 00:40:21,200
kind i offered you
guys before.

845
00:40:21,200 --> 00:40:24,725
And that's inconsistent with
these preferences.

846
00:40:24,725 --> 00:40:28,410
The third explanation
is entertainment.

847
00:40:31,360 --> 00:40:35,110
It's that the utility function
has in it the

848
00:40:35,110 --> 00:40:37,510
thrill of the risk.

849
00:40:37,510 --> 00:40:39,600
We only write down utility
functions that are a function

850
00:40:39,600 --> 00:40:41,980
of consumption like how many
pizza and movies you see.

851
00:40:41,980 --> 00:40:44,780
But people have utility
over lots of things.

852
00:40:44,780 --> 00:40:46,770
One thing you may have utility
of the thrill of being able to

853
00:40:46,770 --> 00:40:50,010
scratch the thing off and seeing
if they won or not.

854
00:40:50,010 --> 00:40:53,580
That would actually be
consistent with the fact that

855
00:40:53,580 --> 00:40:55,100
people play a lot of
small lotteries.

856
00:40:55,100 --> 00:40:57,540
If it's a thrill of winning
that matters, if it's the

857
00:40:57,540 --> 00:40:59,560
scratch off thrill that matters,
then the optimal

858
00:40:59,560 --> 00:41:02,480
thing to do, in fact, would be
to not play one Mega Million.

859
00:41:02,480 --> 00:41:04,530
It would be to play lots
of little lotteries.

860
00:41:04,530 --> 00:41:06,490
And that would be consistent
with that behavior.

861
00:41:06,490 --> 00:41:08,550
So one story that is consistent
with what we see is

862
00:41:08,550 --> 00:41:10,930
that people actually view
this as entertainment.

863
00:41:10,930 --> 00:41:12,450
On the other hand, once again,
it's really expensive

864
00:41:12,450 --> 00:41:13,220
entertainment.

865
00:41:13,220 --> 00:41:17,380
Because you're throwing away
$0.50 of every $1.

866
00:41:17,380 --> 00:41:18,950
So you've got to get a lot of
enjoyment out of that scratch

867
00:41:18,950 --> 00:41:22,060
off relative to when you
go to see a movie.

868
00:41:22,060 --> 00:41:23,310
So that's another theory.

869
00:41:36,380 --> 00:41:37,260
I'm going to put this in here.

870
00:41:37,260 --> 00:41:39,080
It sort of inserts in here.

871
00:41:39,080 --> 00:41:42,330
We talked about the fact that
people can't be risk-loving

872
00:41:42,330 --> 00:41:45,220
because they buy insurance.

873
00:41:45,220 --> 00:41:46,840
And this alternating thing
doesn't work, because they

874
00:41:46,840 --> 00:41:48,500
play small lotteries.

875
00:41:48,500 --> 00:41:51,650
But another theory that might
fit here is a theory we call

876
00:41:51,650 --> 00:41:53,400
loss aversion.

877
00:41:53,400 --> 00:41:55,990
This is sort of a different
version of the Friedman-Savage

878
00:41:55,990 --> 00:41:58,100
preferences.

879
00:41:58,100 --> 00:42:01,235
It's that people are, in
general, risk averse.

880
00:42:03,850 --> 00:42:08,120
But, in fact, they're really
risk averse on the downside,

881
00:42:08,120 --> 00:42:10,850
and they don't care so
much on the upside.

882
00:42:10,850 --> 00:42:14,200
So, in other words, the point
is that when I initially

883
00:42:14,200 --> 00:42:20,040
offered you that bet of win
$125, lose $100, part of your

884
00:42:20,040 --> 00:42:21,950
reaction was about the
risk aversion.

885
00:42:21,950 --> 00:42:23,470
But a lot of you are thinking,
I'd be really

886
00:42:23,470 --> 00:42:25,170
bummed if I lost $100.

887
00:42:25,170 --> 00:42:26,160
It's not just that I don't
have it to spare.

888
00:42:26,160 --> 00:42:28,700
It's just like, god, I
would kick myself.

889
00:42:28,700 --> 00:42:29,890
It was one flip of the coin.

890
00:42:29,890 --> 00:42:32,990
How could I possibly have
been so stupid?

891
00:42:32,990 --> 00:42:34,330
Whereas if you won,
you'd be happy.

892
00:42:34,330 --> 00:42:37,010
But then you'd go on
to the next class.

893
00:42:37,010 --> 00:42:39,860
The notion is that basically
it's an extreme version of

894
00:42:39,860 --> 00:42:41,480
risk aversion.

895
00:42:41,480 --> 00:42:42,850
It's not only that you're
risk averse, it go

896
00:42:42,850 --> 00:42:43,760
further than that.

897
00:42:43,760 --> 00:42:47,150
Relative to the starting point,
anything which is a

898
00:42:47,150 --> 00:42:49,670
loss really pisses you off.

899
00:42:49,670 --> 00:42:53,860
So, in fact, even that little
gamble I offered you, win

900
00:42:53,860 --> 00:42:56,560
$1.25 lose $1, you still
might not take.

901
00:42:56,560 --> 00:42:58,850
Some of you still wouldn't
take it.

902
00:42:58,850 --> 00:43:00,130
And the reason you
wouldn't take it

903
00:43:00,130 --> 00:43:01,635
can't be risk aversion.

904
00:43:01,635 --> 00:43:03,680
Because it's just too small
for risk aversion

905
00:43:03,680 --> 00:43:04,270
to plausibly work.

906
00:43:04,270 --> 00:43:06,505
It's that you'll just be bummed
that you did that and

907
00:43:06,505 --> 00:43:07,710
you took that chance.

908
00:43:07,710 --> 00:43:10,520
You'd be made sadder by the
loss than you'd be made

909
00:43:10,520 --> 00:43:13,520
happier by the win.

910
00:43:13,520 --> 00:43:17,740
In that case, that could explain
why people spend a lot

911
00:43:17,740 --> 00:43:19,110
of money to buy insurance.

912
00:43:19,110 --> 00:43:23,100
Because they'll be so bummed
if things go badly.

913
00:43:23,100 --> 00:43:28,230
But they might play the lottery
because, in fact,

914
00:43:28,230 --> 00:43:30,760
around that point, they don't
view the money they're

915
00:43:30,760 --> 00:43:32,040
spending as a loss.

916
00:43:32,040 --> 00:43:32,710
They think of it differently.

917
00:43:32,710 --> 00:43:36,280
They think of the loss of being
my house burned down.

918
00:43:36,280 --> 00:43:36,910
That's a loss.

919
00:43:36,910 --> 00:43:37,650
That would make me really sad.

920
00:43:37,650 --> 00:43:39,860
But the $1 I paid to pay
the lottery, that's

921
00:43:39,860 --> 00:43:41,860
not really a loss.

922
00:43:41,860 --> 00:43:45,520
So I'm risk neutral going
up and really risk

923
00:43:45,520 --> 00:43:46,910
averse going down.

924
00:43:46,910 --> 00:43:49,330
So I'm willing to take gambles
that push me up.

925
00:43:49,330 --> 00:43:52,012
It's sort of like
Friedman-Savage.

926
00:43:52,012 --> 00:43:54,060
I'm willling to take gambles
that push me up, not gambles

927
00:43:54,060 --> 00:43:54,660
that pull me down.

928
00:43:54,660 --> 00:43:58,430
But, once again, that doesn't
really explain the small ones.

929
00:43:58,430 --> 00:44:00,040
That doesn't really explain
the small ones.

930
00:44:00,040 --> 00:44:02,380
That's more the entertainment
theory.

931
00:44:02,380 --> 00:44:05,920
Then finally, the last
theory we have is

932
00:44:05,920 --> 00:44:07,170
that people are stupid.

933
00:44:09,700 --> 00:44:13,750
The lottery is, after all,
its official motto is

934
00:44:13,750 --> 00:44:15,890
a tax on the stupid.

935
00:44:15,890 --> 00:44:16,710
And that's what it is.

936
00:44:16,710 --> 00:44:17,710
It's a tax on the stupid.

937
00:44:17,710 --> 00:44:21,080
Basically many of your public
schools are financed by taxes

938
00:44:21,080 --> 00:44:21,980
paid by stupid people.

939
00:44:21,980 --> 00:44:23,930
It's sort of ironic.

940
00:44:23,930 --> 00:44:27,350
But people just don't know.

941
00:44:27,350 --> 00:44:30,180
You probably all had a vague
sense that the lottery wasn't

942
00:44:30,180 --> 00:44:31,350
a sensible thing to play.

943
00:44:31,350 --> 00:44:32,700
But how many people actually
knew it was that

944
00:44:32,700 --> 00:44:33,710
bad a deal as I said.

945
00:44:33,710 --> 00:44:36,920
That is actually was $0.50
expected payoff.

946
00:44:36,920 --> 00:44:38,160
A few of you knew.

947
00:44:38,160 --> 00:44:41,070
But most of you knewm had a
vague sense it was a bad deal.

948
00:44:41,070 --> 00:44:42,350
You didn't know how
bad a deal it was.

949
00:44:42,350 --> 00:44:43,840
This is sort of hard
to figure out.

950
00:44:43,840 --> 00:44:46,920
Meanwhile, you see on TV that
these guys win these bazillion

951
00:44:46,920 --> 00:44:50,060
dollars, and you get the thrill
of scratching if off.

952
00:44:50,060 --> 00:44:55,035
So, basically, if people are
just stupid, then that could

953
00:44:55,035 --> 00:44:55,300
explain it.

954
00:44:55,300 --> 00:44:58,550
The problem is it matters a
lot for government policy

955
00:44:58,550 --> 00:45:00,640
which of these is right.

956
00:45:00,640 --> 00:45:03,140
Because if A through C is right,
if one through three

957
00:45:03,140 --> 00:45:06,780
are right, then the government
should go

958
00:45:06,780 --> 00:45:07,880
ahead and allow lotteries.

959
00:45:07,880 --> 00:45:10,490
And there's no reason why the
state shouldn't run a lottery.

960
00:45:10,490 --> 00:45:12,170
In fact, let's take the
entertainment theory.

961
00:45:12,170 --> 00:45:14,260
If this is really entertainment,
and the state

962
00:45:14,260 --> 00:45:15,800
can make money off of
my entertainment,

963
00:45:15,800 --> 00:45:17,920
then that's a win-win.

964
00:45:17,920 --> 00:45:19,750
I'm happy, because I'm
playing the lottery.

965
00:45:19,750 --> 00:45:21,960
The state is happy, because
it's financing schools.

966
00:45:21,960 --> 00:45:23,220
That's a win-win.

967
00:45:23,220 --> 00:45:27,420
So if these are right, you're
going to want to encourage

968
00:45:27,420 --> 00:45:29,090
state lotteries.

969
00:45:29,090 --> 00:45:31,760
But if this one's right, we
don't want to have them.

970
00:45:31,760 --> 00:45:35,365
Because, A terrible way to raise
government revenues is

971
00:45:35,365 --> 00:45:37,740
to tax stupid people.

972
00:45:37,740 --> 00:45:39,310
There are much better ways to
raise government revenues.

973
00:45:39,310 --> 00:45:41,240
We'll talk about taxation
in a couple of lectures.

974
00:45:41,240 --> 00:45:43,270
But, clearly, taxing the stupid
is not going to be an

975
00:45:43,270 --> 00:45:43,980
optimal tax.

976
00:45:43,980 --> 00:45:45,018
Yeah.

977
00:45:45,018 --> 00:45:47,826
AUDIENCE: I can maybe sort of
understand why people would

978
00:45:47,826 --> 00:45:50,166
prefer smaller lotteries
over bigger lotteries.

979
00:45:50,166 --> 00:45:53,050
Because they are thinking that
in smaller lotteries, they

980
00:45:53,050 --> 00:45:54,953
have a much bigger
chance of winning

981
00:45:54,953 --> 00:45:55,780
than in bigger lotteries.

982
00:45:55,780 --> 00:46:00,785
So, in that sense, their
expected payoff in terms of

983
00:46:00,785 --> 00:46:03,719
utility or other
[INAUDIBLE PHRASE]

984
00:46:03,719 --> 00:46:07,435
is a lot higher than the antes
in the bigger ones, even

985
00:46:07,435 --> 00:46:11,450
though the bigger ones might end
up being a lot heavier--

986
00:46:11,450 --> 00:46:13,100
PROFESSOR: So that's sort of an
entertainment theory, which

987
00:46:13,100 --> 00:46:16,260
is my utility derives
from the win.

988
00:46:16,260 --> 00:46:19,430
You have a theory in mind my
utility derives from the win.

989
00:46:19,430 --> 00:46:21,710
Because if it's just
about dollars, that

990
00:46:21,710 --> 00:46:22,430
wouldn't explain it.

991
00:46:22,430 --> 00:46:25,080
Because I win so many more from
the big one that it would

992
00:46:25,080 --> 00:46:26,870
compensate from the frequency
at which I'd

993
00:46:26,870 --> 00:46:27,760
win the little one.

994
00:46:27,760 --> 00:46:30,430
But if I actually, in my utility
function, have the joy

995
00:46:30,430 --> 00:46:33,240
of seeing that winning thing,
then that would explain it.

996
00:46:33,240 --> 00:46:34,540
That's an entertainment
theory.

997
00:46:34,540 --> 00:46:36,790
You're saying, in my utility
function, I actually get joy

998
00:46:36,790 --> 00:46:38,430
from scratching off and seeing
that it's a winner, and so

999
00:46:38,430 --> 00:46:43,070
much joy that I'd much rather
take a 10% chance at a small

1000
00:46:43,070 --> 00:46:45,180
win than a 1% chance
at a huge win.

1001
00:46:45,180 --> 00:46:47,800
Because then, at least, with the
first one, 1 in 10 times I

1002
00:46:47,800 --> 00:46:49,990
get that joy of the scratch
off and seeing it's a win.

1003
00:46:49,990 --> 00:46:51,300
So that's sort of
an explanation.

1004
00:46:51,300 --> 00:46:55,820
And that would say that
lotteries are good.

1005
00:46:55,820 --> 00:46:57,550
The other way economists might
think about lotteries is

1006
00:46:57,550 --> 00:46:59,320
they're voluntary taxes.

1007
00:46:59,320 --> 00:47:02,000
The public doesn't like taxes.

1008
00:47:02,000 --> 00:47:04,870
Here's a voluntary tax.

1009
00:47:04,870 --> 00:47:08,530
You never hear policy makers
getting up and railing against

1010
00:47:08,530 --> 00:47:11,280
a horrible evils
of the lottery.

1011
00:47:11,280 --> 00:47:13,000
Sometimes groups do.

1012
00:47:13,000 --> 00:47:15,670
Sometimes outside groups
do and stuff.

1013
00:47:15,670 --> 00:47:16,760
But politicians don't.

1014
00:47:16,760 --> 00:47:18,940
But those same politicians will
go on and on about how

1015
00:47:18,940 --> 00:47:19,860
terrible taxes are.

1016
00:47:19,860 --> 00:47:20,220
I'm going to cut your taxes.

1017
00:47:20,220 --> 00:47:21,875
Taxes are terrible.

1018
00:47:21,875 --> 00:47:24,460
Well, the lottery is a voluntary
tax in that sense.

1019
00:47:24,460 --> 00:47:26,210
And I might say, look, there's
no reason to oppose it, it's a

1020
00:47:26,210 --> 00:47:27,130
voluntary tax.

1021
00:47:27,130 --> 00:47:28,200
It's those involuntary
taxes that

1022
00:47:28,200 --> 00:47:31,180
cause problems in society.

1023
00:47:31,180 --> 00:47:34,080
Well, whether we want to buy
that story or not depends on

1024
00:47:34,080 --> 00:47:36,710
how much we think it's being
played because people are

1025
00:47:36,710 --> 00:47:37,305
stupid or not.

1026
00:47:37,305 --> 00:47:37,560
OK.

1027
00:47:37,560 --> 00:47:39,120
Let me stop there.

1028
00:47:39,120 --> 00:47:41,500
So that's a great example of
how a little bit of an

1029
00:47:41,500 --> 00:47:44,760
extension of our model
can really enrich our

1030
00:47:44,760 --> 00:47:47,070
understanding about a lot of
decisions that we make in the

1031
00:47:47,070 --> 00:47:48,000
real world.

1032
00:47:48,000 --> 00:47:49,175
We'll come back and
talk about another

1033
00:47:49,175 --> 00:47:53,500
version like that later.

1034
00:47:53,500 --> 00:47:55,650
And that is the case of
thinking about savings

1035
00:47:55,650 --> 00:47:58,480
decisions and thinking about
individual decisions on how

1036
00:47:58,480 --> 00:47:59,980
much to save and how
much to spend.