Okay, great, so last class we finished up on, we put together
the foundations on rational behavior, transitivity and
completeness, and what we're trying
to do now is build a framework for
us to understand preferences and how
to link it to utility. So we spoke about
this idea of expected value,
and then we showed how expected
value on its own isn't enough to capture
how people should behave. because firstly
risk preferences matter we saw some people in
this class were risk loving wanted to take
more risk this is why people bet on
something like roulette i don't know if we've
discussed the probabilities of roulette in this
class um has anyone here played roulette
before any chance at the casino anyone we're
all we're all good people you know not
going to the casino but does someone know if you
bet a dollar on black and it lands on black
how much do you win You get a dollar, yeah. If you lose, you
lose a dollar. But what's the
probability of the ball landing on
black in a roulette? Yeah, so it's not 50%.
It's less than 50%. They put either one,
two, or three greens on the wheel as well
as red and black. And this means
this is a negative expected value bet.
For each dollar you put in, you're going
to lose more than a dollar on average.
But still, a ton of people, no
one in this class, a ton of people go
and play roulette. I remember when I
was younger and I went to the casino,
there was all these people that looked
like they had this amazing system,
and back then they were like, oh my god,
they're so smart. Now I'm like, no,
they're just idiots. This is negative
expected value. So, people do that
because they're risk -loving, and a lot
of us will take risk -averse behaviours
as well, like taking out car insurance
or house insurance. These are risk-averse
styles of behaviour. The other thing we
talked about, and we demonstrated as well,
thanks to our volunteers, this idea of
diminishing returns. He's one of them right
now. He's Christian. So diminishing
marginal utility, the idea that the next
dollar gives you slightly less happiness than
the previous. And diminishing utility and
diminishing returns, as we saw, go to a
lot of behaviours. And just to drive
this point home one more time, I actually
want to do another activity where I need
one more volunteer, some caveats to this
volunteer, I realized the one I did on Wednesday
was probably more male-coded, so I want
a non-male volunteer. There's nothing bad
that can happen. You don't have to do
anything. You've just got to make a few little
decisions, and there can only be good things
that come from it. So before I pick
someone out, does anyone that's not a male
want to volunteer? Not a male. Yeah, yeah. Anyone? Yeah, Brooke, come on up. Okay, Brooke. Look, so you're going
to play this game with me for real
money, real money. You can't lose anything,
you can only gain. So the way it works
is there's going to be a grid of 5x5 boxes
labelled 1 to 25. And behind 24 of these
boxes is 50 cents. And one at a time you
can open a box and you can see if there's
50 cents inside. And at any point
you can stop and collect your winners. And I'll pay you
those winners. However, behind one of
these boxes is a bomb. and if you select
a bomb at any point before collecting your
winnings you get nothing understand the rules
great so here we go do you have a lucky
number that you want to start with 13 okay
really lucky 13 so it's Friday the 13th today
that's kind of ironic um Brooke do you know
what the probability of this box being a bomb
is by any chance yeah 4% yeah okay so let's
see if you get the coin nice good for you
all right and remember you can collect it
anytime anytime one nice three no does anyone
know what this sounds from by the way Mario
yes the Mario coin sound 21 21 why 21 fair
enough any desire to check out already or
you want to keep going Really one walk I call Okay, do you
want to change she check out
I'll keep going They go The crowd don't shoot
you do heroin as well everyone. It's
like you know peer pressure. Do what
you want to do Brooke There's one bomb
one bomb six Are you sure you
don't you said you wanted to check
out are you sure? Nice I'll do seven, just
for... Seven. You're gonna stop two ago
now. I'll start after seven. After seven,
okay. Are you sure? Yeah. Okay, let's
click seven. You checking
out? Checking out? Congratulations. The bomb was behind 20.
I'm gonna write this down. Email me your
Zelle or Venmo, whatever you have, and
I'll get you the money. Well done, Brooke! Because Brooke
checked out relatively early, I'm gonna get
one more volunteer up here. no no restrictions
on oh yeah okay Brooke so $3.50 okay
Tess do you think you're more risk
loving than Brooke all that okay yeah I
mean you're a Packers fan so like hyper
offense at the moment okay what are you
gonna start with 20 20. Nice. 21. 21. Uh, 7. Why? Why the
switch to the 7? Just random. No worries. Uh, 6. 6. What was the, what
was the, there's no 30. It goes up to 25. 3. 3. Why, why, who
said whoa and why? No, no, no, no,
no. Why? I mean, I think she tied up
for the last one today. Okay, yeah. So
if it was you, would you stop right now? Okay, you keep going.
Okay, number 3? Yeah. Commentaries allowed
say your thoughts 19 you sure 19 all right should test keep going Okay, at least
till five okay That's my lucky number 25 What what who
said knock out the corners why the
corners fair enough? Do you want to
knock out the other corner do you
want to stop okay? Where's your head at? You've got $6
.50 right now. I don't know if
you're over 21, but this is a drink at
Harry's, if you are. I'm not. Well,
then it's a coffee. Let's do 13. Okay, 13. Unlucky 13. Remember, if it's a bomb,
you get nothing. Bad life test. Okay. Okay. I'm sad that you
didn't get money, but I'm glad my little
video got played when you hit the bomb.
But thanks for playing, Tess. Thanks
for playing, Brooke. And as per usual, I
like to do fun things, but I don't do
them just because they're like fun. Like
we tie them back to something. So as we
talked about before, picking any box on the
first time, you have a 4% chance of hitting
the bomb. so essentially what we can do is
we can calculate the expected value of opening
boxes and by people's decisions here we can
look at what Brooke did, we can look at
what Tess did and think about what you would
do and we can actually see if you're revealing
yourself to be risk-averse, risk
-neutral or risk-loving so before I get to it does
anyone know what the highest expected value
is in terms of which one gives you the most
expected value so if we go down we can see
if you open the first box there's a 96%
chance you get the coin 50 cents and a 4% chance
you get nothing so if we multiply 96 by 0
.5 and 0 by 0.4 we get the expected value
here which is 0.48 and let's say your original
plan is to open two boxes then there's an
eight percent chance that one of the boxes
was a bomb and there's a 92 percent chance
there's no bomb and you get a dollar and we can
calculate the expected value for any plan
of opening boxes as follows and as you can
see the expected value is increasing
increasing increasing up until 12 and 13 boxes
that's when expected value is maximized and
after that, as you can see, it's decreasing,
decreasing, and decreasing. So expected value
is maximized when you pick 12 or 13 boxes.
And you can see this is the case with the
marginal expected value of opening the box.
How much do you get on average by opening the
next box? So you can see up until 13, it's
positive, decreasing, but positive, and then
it becomes negative after that. So, Brooke,
you stopped at 7? So this is risk
-averse behavior. So on the margin,
in terms of expected value, you could have
got more out of it. The bad outcome here
is pretty bad, isn't it? Tess knows from
first-hand experience. So most people
actually exhibit risk-averse behavior, and this is how we
can measure it. Tess, on the other
hand, at what point did it
blow up for you? I think it was at 650. 650. Okay, so, yeah, you
were at 650, then you went to $14. So this
was the first negative expected value guess
when it's more likely to blow up essentially
than not based on what you can gain from it.
But there's nothing wrong with that as
well. As we said before, you actually said, I'm
much more risk loving than Brooke and that's
how it played out. You were going for the
big prize essentially. Like for you, yeah,
like whatever, $6.50 would be nice, but
hey, I wanted like $10 or something like
that so this is a good example a what we
actually use we use this actually up in the
lab with less um graphics and videos
and stuff to measure people's appetite for
risk as you can see if we go back here if
we i have to go back to my history or
whatever yes this is actually called the bomb
risk elicitation task developed by crescento
and filipini in 2013 this is just one
of many ways we try and measure people's
appetite for risk. Great. So thanks to our
two volunteers and we're going to
bring this back. So the whole reason we have
this idea of expected utility instead of
expected value is because people aren't expected
value maximizers nor normatively should
they be. People's appetite for risk matters
and we want to take into account this idea
of diminishing marginal utility. Think about
where you would have stopped at but I
feel like 80% or more people in this class
would have stopped it somewhere except for 13
and 14 in terms of the boxes, showing that
they have different risk preferences to the
expected value model. So if you remember,
once we put in these assumptions
of rational choice and we add a few
other things, we can link people's
preferences to utility. This is exactly what
we're going to do in this class. We want
to be able to say the utility someone gets
from option X is greater than or equal
to the utility they get from option Y, and
this means they must weakly prefer x to
y. If the utility number is high for x
and y they must strictly prefer x to y and
if it's equal that means they're indifferent
between x and y. So what is a utility
function doing? What can they do and what
can't they do? This is kind of controversial
especially at the moment there's a lot
of debate going on. So the utility function
is just a scale to rank different
alternatives. The numbers themselves don't have
any cardinal meaning. So we talk about
ordinal and cardinal. So cardinal is
like the magnitude of the difference
between numbers. So the difference
between 7 and 10 is 3. This 3 matters
in some respect. However when it's
ordinal we just care about the rank in the
highest number to the lowest number. So
in 10 and 7 the 3 difference is meaningless.
All that matters is that 10 is higher
than 7. So a utility function can tell
us that a person prefers apples to
oranges. What they can't tell us is that a
person prefers apples by 73 utils to oranges.
That's a meaningless statement in expected
utility theory. So for example, if
the utility of banana equals 1 and the
utility of an apple equals 2, we can
say that because the utility of the apple
is strictly greater than the utility
of the banana, this person prefers the
apple to the banana. So what it's
essentially doing is it maps the set of
preferences, the choice alternatives, into
the set of real numbers. And this
is what represents this preference
relation, if the following is true.
So as we said before, the utility of A
is greater than or equal to the utility
of B, if and only if necessary and
sufficient conditions if A is at least
preferred as to B for all A and B
in the choice set. As we said before,
what utility functions cannot do is if the
utility of a banana is equal to 1 and the
utility of the pear is equal to 3, we
can't say this person prefers pears three
times more than bananas. We can't say they get two more utils
from it either. All we can say is that
they prefer pears to bananas so the actual
difference in numbers carries no cardinal
meaning it's just the order that matters
okay has anyone here seen the movie oppenheimer
by any chance anyone a few people so von
neumann here makes a brief appearance and
if you went on to any reddit threads after
you'd have a bunch of nerds like me being
angry that von neumann wasn't more prevalent
in the movie because he was actually like
foundational to the development of the nuke.
He was like arguably just as important as
Oppenheimer himself. And von Neumann is
like very prolific. He also started game
theory so we'll come back to him later
but he along with Morgenstern axiomized
expected utility theory. They essentially
discovered how to have these utility
functions and map preferences into these
utility functions. So they say utility
theory can work, we can represent preferences
with a utility function, they satisfy
the following axioms. Completeness, we've already discussed
this one. Transitivity, we've
discussed as well, this is our backbone
of rational behavior, and then they add a
couple more things. Continuity, this is
just beyond this course, I'm not going to go
into the definition of it at all, but this
is just to show you can have a continuous
essential utility function that doesn't
jump up and down at all. Something called
independence and kind of like the sister of
independence is something called the independence
of irrelevant alternatives and if
you remember the only difference between
expected utility and expected value is we're
taking the outcome and we're transforming
it into utility and the reason why we're
transforming it is because ten dollars
might not always be worth ten dollars
you've got zero dollars ten dollars might be
worth a lot you've got a billion dollars
ten dollars probably isn't worth that much
in terms of utility. So that's why we
transform it before multiplying the outcome
by the probability. Alright, so independence.
I'm going to give you a very
technical definition. You don't need to
remember, and then I'll give you a more
intuitive example. So don't be afraid about what you're about to see. Okay, so what this says is, if you prefer x to y, x has a 100% chance
of happening, y has a 100% chance
of happening, and you prefer x to y, Then,
if we add a third option to the mix,
and reduce the probability of X and Y
happening by the same amount, so let's say
X has an 80% chance of happening, Y also
has an 80% chance of happening, and we
add this third option with 20% chance of
happening to each, then you prefer X
with 80% plus Z with 20% to Y with 80%
plus Z with 20%. It's the technical definition
of independence. But I've got a
more intuitive example to help
you out here. Okay. Right. What's your name
here? Kevin. What do you prefer? The apple or the banana? Apple. Apple. You
a big apple guy or just... Always
go with an apple. Always go with
an apple. Great. Okay. How about
this? I'm going to offer you a lottery
where you have a 50% chance of getting
an apple, 50% chance of getting $1, or
your other option is the 50% chance
of getting a banana. and a 50% chance
of getting $1. What do you prefer? I'll go with that. Okay. What's your
reasoning for it? Without knowing any
of the technical details, what's
your intuition here? I would say because
they equally... I still get the same
chance to get money. Yeah. Like money
works, then foreign foreign actions. Exactly, exactly. So
nothing's actually changed in these
decisions, or at least the difference between
these decisions. In both cases,
you have a 50 % chance to get
the dollar. So that doesn't make
any difference between these two options. And
because you've already said you prefer the
apple to the banana, nothing has changed
between it. So if you prefer the
apple to the banana here, by reducing
each of these choices by the same
probability, and adding a third option that's
the same in both choices with the
same probability, you need to stick,
or not need to, you should stick
to your original choice if you
want to, you know, follow this axiom
of independence. Does that make more
sense to people? Great. We have another
one called the independence of
irrelevant alternatives. So independence
is more for probabilities and
lotteries. This is just for
straight-up choices. So if I offered Kevin the apple and the banana, he clearly
chooses the apple. So if I offered Kevin
a different choice, the apple, a banana,
or the orange, So I've added this third
alternative, the orange. Would it be
weird if you now pick the banana
over the apple? Yeah, like that
doesn't make any sense. Why would you do
that? This is what the independence
of irrelevant alternatives says. A
new option shouldn't flip how you rank
two existing options. So if you prefer
X to Y when only X and Y
are available, then you need to prefer
X to Y in a choice set that has both X
and Y available, and we added an irrelevant
option. z your ranking of x and y
shouldn't change it's independent to this
irrelevant alternative that we're throwing in
and this is important because choice is a
context independent it avoids manipulation
of preferences and it simplifies
economic models and as i said before
this is kind of distinct from the
independence action independence is for
lotteries, probabilities, there's no
probabilities here. And while this is
like not part of this course, if you
take my behavioural course in the
future or with Professor Kovac or
Professor Duffenberg, it's a bunch of
us teaching it, what we essentially
show is there's a tonne of people
violating a maxim like this, which Kevin,
that seems kind of crazy to you, violating
this, correct? Yeah. But there are
ways, and I'll give you an example now.
So let's see if I can get a popcorn size
prices let's see if we have an image
here great oh this is perfect all right
great so imagine you're in a cinema and they
only have a small and large there's no
medium yeah so you can spend four dollars
for the small or you can spend seven
dollars for the large so there's kind
of a trade-off here. The large is
more expensive, but it's also bigger. In this situation,
let's say a person chooses a small. Next, you introduce
the medium. So now you've got three
options. Remember, this person chose
a small originally. Now, when evaluating
these choices, this person looks at
the large, then looks at the medium,
and it's like, oh, the large is much
better value it's only 50 more cents and
you get like twice the amount of popcorn
whereas you know the small just doesn't
seem like great value anymore so
what this has done is we call this the
decoy effect this irrelevant option
has made one of the other options look
better by comparison as a result someone
who originally would have chosen small now
might flip to choosing large instead and
as some of my other students in the
previous class they said oh yeah this is
essentially what mogging is when you're able to
make yourself look better than someone
else so if by comparison it changes the way
people feel about you in terms of looks i
guess so this is one way iira can be
violated finally not an explicit axiom but
violating this would also violate expected
utility theory in variance a decision maker
should not be affected by the way alternatives
are presented So the ranking of
preferences should not depend on the description
of the options. This is description
invariance, which I care more about you knowing.
The other one is more experimental
-based, so don't worry about it. But it's the
method of elicitation, procedural invariance.
And I can give you an example of that if
you are interested. But for description
invariance, imagine you
have an option between a Danish
and a doorstop. I feel like most
people choose the Danish over a
doorstop if it's all not eating
it. But as you can see in this famous
Simpsons scene, Bart says, Dad, I'll
trade you this delicious store stock for your
crummy old Danish. And in the scene,
Homer makes the trade because he succumbed
to Bart's marketing the way he described
each of the objects. So just by marketing
and describing it differently, it flips
Homer's preferences so he's violating
invariance here. So if anyone here is a
marketing major, this is one of the aspects
of marketing that you can change people's
tastes and get them to actually violate
this idea of invariance. finally to build what
we're going to look at next our indifference
curves we need to add a little bit more
structure because currently it could still
potentially be you know um it's still rational
but we want to get rid of these ideas of
being able to prefer less to more our good
old friend bob who wants to always take the lowest
paying job possible we don't want that
so we added a little bit more structure to
our expected utility theory so we're able to
construct in difference curves which is what
we're looking at next so we already
looked at this idea that utility is not calculated
in the abstract ten dollars is not ten
dollars ten dollars is in utility it depends
on do you have zero dollars do you have a
billion dollars you have a thousand dollars
that's what you calculate utility over total
wealth non satiation is really important
here you can never have enough utility you
can never have enough money there's no awkward
limit to happiness and while you know you
might question this assumption i mean i
would counter that by pointing out you know
some of the richest people in the world right
now are always seeking more money more power
etc so i feel like non -satiation is actually
a pretty good axiom monotonicity more
of a good thing is always better you'd
always prefer an extra dollar so this
counts out good old bob who wants less
money we don't have that anymore unexpected
utility theory. Convexity, which I'll
discuss later. And finally, this idea of
diminishing marginal utility, which
I've talked about a lot. The next dollar
gives you slightly less happiness than
the previous dollar. So this is called an
indifference curve, and this is why I
spent an election and a half leading up
to this. This can be misunderstood so
much, but you'll understand what is going
on here now. So far, we've just been ranking
individual things. So one apple versus
one orange etc. But a lot of decisions
involve multiple goods or multiple
dimensions and I'll get to a few examples in a
second. And what an indifference curve
does is show you what combinations of things
give you the same amount of utility, the same
amount of happiness. So along an
indifference curve here as you can see
there are different bundles of good
x and good y. At point A, you get
6 of y and 1 of x. At B, you get 4
of y and 2 of x. And at every point
along this line, it gives you the
same utility. Hence why they're
called indifference curves. A person
is indifferent between any point
along these curves. Alright, so as we just
talked about, this can be with goods. You
could have like 7 apples and 3 oranges or 4
apples and five oranges how do you compare
each of these bundles what about attributes
finding a partner you might care about
personality you might care about looks how do you
trade these two things off this is something
we kind of talk about a lot in the abstract
what about choosing the college when you
chose Purdue I assume you added up a whole
bunch of reasons for coming here we're
evaluating other colleges two of the more prominent
ones are probably the academic rigor the
value of the degree here also just you know
campus life in generally you're spending four
years here in the peak of your youth you
want to pick a place where you're going
to enjoy so different combinations mean there
are many different points of indifference
there's many different points where you
increase one decrease the other where you'd be
just as happy for example you could be indifferent
between seven apples and one orange
and two apples and three oranges or if you rank
them out of ten you might enjoy a university
that has a nine campus life in a three
in academic rigor, Arizona State University
for example, or a six in campus life in a
four in academic rigor. I don't know, what's a university
like, IU let's say. So how can we represent
these preferences? We have our good
old indifference curves here and I've
drawn it, well not drawn it, I've used
this image here to show you it's
not just one indifference curve,
it's in fact infinite. They can go all
the way up like this and all the
way down like this. This is what
they look like. And there are four
principles when it comes to indifference
curves. We've kind of alluded to a lot
of them already, so they should come a
bit easier. But the first is, along any
single curve, you're indifferent between
any of the options. The higher the
curve, so up and to the right,
means you get more utility from that
entire curve. Indifference curves
cannot intersect with each other
and finally because of convexity
indifference curves are downward sloping
which gives us this idea of diminishing
marginal utility all right so as we
said indifference along a curve so seven of
good y and three of good x gives you the same
utility as two of good y and eight of good x
in this example here you're indifferent
between these two options i don't have to spend
more time on that one we're all good with the
idea of indifference the higher curve means
higher utility so as you can see as we
move up and to the right from i1 to i2 to i3
you get more utility and it is a simple
reason for this so if we go back here let's say
we're at this point on i1 and we move
horizontally from i1 to i2 does anything does
anyone notice anything about this point on i2
compared to this point on i1 yeah eric yeah
exactly so you've got the exact same amount
of good y but you've got like extra good
x you've probably got what like one or one
and a half more of good x and as we established
before with these auxiliary axioms
monotonicity means more is better than less so this
has to give you more utility because you've
got the exact same amount of y but more
of x so that's why we draw it in this way
up and to the right it would be the same thing
if we looked at you know this point here on i1
and went vertically up to i2 you get the
same amount of good x and you get more of
good y so that's why they're shaped this
way and if you draw it even thinly up like
that you can still prove the same thing
so anything that's above the line in
this black area in the upper contour set as
you say is the preferred area you get more
utility here and you get less utility
anywhere below the indifference curve in
the lower contour set. Indifference curves
cannot intersect. This one's a
little bit harder. Does anyone have
any intuition what might happen if
they intersect? This is way beyond
me when I was a student. Brooke,
give it a crack. Exactly right. just
you were looking at um this before just here
yeah the this yeah yeah that's right i'll get
back to that example in a second so remind
me so brooks 100 correct the basic idea
though is it violates transitivity so if we
look here on indifference curve actually before
we get to that i'm going to do something
here called a proof by contradiction so i'm
going to prove one thing by using
transitivity, and then I'm going to prove
another thing through monotonicity
and show you that they both
can't work together. All right. So on indifference
curve one, we see the points A and
C lie on the same curve. So we must be
indifferent between A and C. They lie
on the same curve. On the IC2, we see
B and C lie on the same curve. So we
must be indifferent. Thus, by
transitivity, we must be indifferent
between A and B. must be indifferent
between a and b however we also know from
before if you have more of each good then
you must prefer the one that dominates the
other so if we look at b you get some
amount of good y and x and as eric pointed
out before if you go up to a you get more
of good y and more of good x so it must
be preferred you're getting more of both
monotonicity says you must prefer a to b so we
have in this situation where the curves
intersect we have a is greater than b and
we also have a and b going different
between them we can't have both those things
this is a proof by contradiction another
simple way to look at it as as as brooke
said before as you can see if we look at
indifference curve one everything in this upper
region is preferred so everything in the
upper region here is preferred so if there's
a point here d this would be in the preferred
region and everything in the red region
is less preferred so you get less utility
from it and as you can see on b this is in
the less preferred region underneath so you
have some things that are less preferred and
some things that are more preferred on ic2
so you're not actually indifferent between
those two points on the curve so there's
a couple of different ways you can prove
why these indifference curves can't intersect
and you're way beyond me when i was at your
stage brooke well done finally this is
slightly more technical the idea of convexity
and downward sloping so this is what a
convex shape looks like if it was concave it
would be drawn like this it would have the
opposite style of arch so this idea of the
convexity and the downward slope in nature ensures
that on a singly indifference curve to
gain more of one good you have to give up
some of the other no free lunches here if you
look here if we start at 10 and 1 to get an
extra good of x the only way to get one
more unit of x is to give up three of good y
to get another unit of good x you've got to
give up two units of good y to give up to get
another good x you've got to give up one
unit of good y, then 0.5. So you're always
giving up y to get x. Or to get more y,
you've got to give up x. And here we have
something called the diminishing marginal
rate of substitution. And what this means,
this is really important, is essentially,
remember, if you've got a lot of one
thing, then getting an extra unit of it
probably won't give you much happiness.
And if you've got little of another thing,
getting one unit of it will give you a
lot more happiness. So as you can see here
at this point here at 10 1 we have a lot
of y and like none of x so here this person
is willing to give up three units of good
y to get one unit of x they're willing to
give up a lot of y to get one x however for
the next unit because they now have slightly
more x and slightly less y they're now
only willing to give up two units of y for
one x then only one unit of y for one x
and now you've got the same amount of x as y
so when you give up um when you get one more
unit of x you're only willing to give up
half unit of good y you value y more now
because it's more scarce so as you move from
a lot of one thing to less of it you're
willing to give up less and less for one unit
of the other good so here are some examples
so this is something you know uh probably
situation many of you will be in on sunday
night it's going to be about like midnight or
1 a.m and you're like oh should I cram an
extra hour for the exam or should I get to sleep
and for all intents and purposes you can
probably do just as well if you get that extra
hour of sleep it's good to be well rested
but in this case here imagine your two choices
are you can have six hours of sleep and
score a 70 on the exam or you can cram an extra
hour get less sleep which kind of sucks
you're going to feel shitty the next day
but you get an extra 10 points on the exam so
the marginal rate of substitution is someone's
willingness to trade off one thing for another
so this student is willing to trade sleep
for points so as you can see they're willing
to lose one hour of sleep if they gain 10
additional points here and we can do the same
thing here this student is willing to lose
one hour of sleep actually this should be
this should probably be be higher not 85 let's
call this 95 instead of 85 as they get less
and less sleep they need more and more on
the exam so this will need to be 15 points
to lose one extra hour of sleep so on our
difference curve the marginal rate of substitution
is just the marginal utility of x how
much of x we gain or lose divided by the
marginal utility of y how much y we gain or lose
so as we move from a to b we move from six
one to two four we gain one unit of x and
we lose two units of y so the marginal rate
of substitution here's going to be one divided
by two a half as we go from b to c you
gain one unit of x and you lose one unit of y
so 1 divided by 1 equals 1 and then if you go
from let's say c to actually no you'll
get to a point this is kind of different
as you can see at any point along
the curve the margin rate of
substitution is changing you will be wanting
to give up less y for 1 extra unit of
x the more you get here so as you can
see from here to here to get 1 extra
unit of x you're probably only giving
up what's that 0.5 units of i, so 1
divided by 0.5 is 2. So the marginal rate
of substitution in this case is increasing
as you go down the curve, as you
go down the curve. Finally a point on this
idea of the marginal rate of substitution
is this principle of convexity and the
reason why I want to bring it up is
people always confuse indifference curves
with utility functions. Utility functions
are concave remember we have that nice
diminution return style our indifference
curves are convex and this is the reason
why so consider two bundles a and b between
which you are indifferent they align the same
curve what convexity says is you'd prefer
to have any convex combination of those
two bundles to either of the bundles themselves
it's pretty wordy convex combination all it
is is a fancy way of saying a weighted average
it's just a straight line between those two
points as i've drawn here not the straightest
line in the world but any point on this
line between a and b you'd prefer to either
a or b so visually preferences are convex
if for any two points of the same indifference
curve a segment connecting those two
points passes through the preferred region so
remember anything above this line is preferred
so this entire line passes through or is on
the preferred region so indifference curves have
this convex property yeah so it's important
to remember utility functions are not convex
indifference curves are so why don't we
have concave curves why do they need to be
convex so a concave indifference curve would
imply as you consume more of one good you'd
be willing to give up more and more of the other
good for an additional unit of the first
so in this case here you start off with
let's say six units of y one unit of x and as
you can see you only start off by being willing
to give up one unit of y for an extra unit
of x and as you get more x you can see
you're actually giving up more and more of y
this is the opposite of diminishing marginal
utility when we go from from three four to five
one you're giving up two units of y even
though you have less y then when you started
here, you're only going to give up one. So
that's why we don't want this idea of concave
utility functions. We want to bake in this
idea of diminishing marginal utility into
our indifference curves. This one's a
little bit more complex. Are we all
still following? Great. So there's a few
special cases of indifference curves
that I want to point out. So D, we have
our normal ones. B are perfect substitutes.
so imagine you're completely indifferent
between coke and pepsi for you it doesn't
matter which one you get they're perfect
substitutes then you're going to get an
indifference curve that is a downward slope in
line with a slope of one so what that means at
any point on this curve you get the same amount
of coke and pepsi added together so
imagine good x is coke and good y is pepsi here
you could have 10 cans of coke zero of pepsi
here you've got five of each here you've
got 10 cans of of pepsi it doesn't matter to
you you always have 10 cans and you value
them the exact same but you still have higher
and lower indifference curves on indifference
curve two you might only get seven seven
and then 3.5 3.5 and here could be um two
two one one something like that the other big
one here are perfect compliments you've got
these weird l shapes so A perfect complement
to something that I would say is a left
shoe and a right shoe. If you have one left
shoe and two right shoes, you're
probably just as happy if you had one left
shoe and one right shoe. That other
right shoe is kind of meaningless. So what
this is saying is, the only way you can
move up and to the right is if you get
one more of each. So what this is saying
is, one of each shoe gives you the same
utility as one left shoe and five right shoes.
or in other words you get more utility from
two of each two left shoes two right shoes
then from one right shoe and five left
shoes or something like that now you can make
arguments saying something like oh you can sell
the extra shoes or if one gets worn out
etc but just yeah like let's just demarcate
this example to at least help the idea
of understanding what perfect complements
are and finally
near substitutes as you can see
they're not as flat as this but here, they're
very close together. So the diminishing marginal
rate of substitution is much lower than
in your normal indifference curves. Like,
even if you've already given up a lot of
1 because you value them close together, it
doesn't change so much along the curve, whereas
the marginal rate of substitution here
changes very quickly. This one is
less important than the other
two, though. So, the most important
thing here is linking this idea of
preferences, which we can show with indifference
curves to utility which we can show
the utility function so as we have said a
few times now we can link preferences to
the utility function in the following way if
the utility of option a is greater than the
utility of option b this must mean that a
is at least as preferred as to b and we can
show it in this way so on the left here we
have our indifference curves downward sloping
they're convex and on the right hand side
we have our utility function so this could
be money it can be anything that's
represented by by these bundles and this is
concave which gives us increasing utility but
it's increasing at a decreasing rate so for
each extra dollar you get you're happier
but it gives you less happiness than the
previous dollar and as you can see on these three
indifference curves I've labelled three
points, A, B, and C, this A could be
anywhere on I1. And every point on
I1 is translated to the utility
function like here. So this point here, if
we called it D, would occur at the same
point of the utility function as A. This
is how we map the indifference curves into
the utility function. For B, which
occurs on I2, which is up and to
the right of I1, this occurs at a
higher point on the utility function, and
the same thing with C. But the thing to keep
in mind is anything on the same curve
will appear at the same point on
the utility function. Does anyone have
any questions about indifference curves?
We're going to be seeing them a lot in
the next class, so it's important that
you understand them. To bring this all
back, I wanted to talk about a few things
regarding utility that matter a lot. Does
utility and this idea of more is better
than less actually play out in the real
world? For example, does more money lead
to more happiness? So I've got a graph
up here, but I've got a bit of time.
I just want to get people's thoughts.
Do people think more money always leads
to more happiness? Yeah? Do you want to...
No, I thought we were... Sorry.
No, no, no. If you don't want
to say anything, you don't have to,
but I'd love to get your thoughts.
Oh, just thanks. Great. Anyone want
to counter that in any way? Can we get
some disagreement? Is that disagreement
or is that a stretch? It's kind of a stretch.
let's say like more money doesn't always
equal more happiness once you get to a certain
point but like you need to have enough
money or you're going to get like necessities
and stuff like that great so there's a
sufficient level of money so you say this idea
of non-satiations like probably not right at
some point we can be satisfied with enough
money and we prefer other things rather than
their money potential so as you can see
here here's a graph of happiness scores across
the world the the bigger the smiley faces
the more happy they are if you've got like
a face like this it's a score between five
and six and the sad faces are a score less
than five and these are on scales of one
to ten but they ask how happy are you and we'll
get back to that so that's one part and
the other part is the darker the green the
higher the gdp per capita the higher the money
so we've got all these things here and
before we get back to that i want to show you
a really cool paper by ricardo peres truglia
called the effects of income transparency
on well-being evidence from natural
experiment. So a natural experiment is when there's
a random law change in the country and
there's differences in behavior because of
that random change which economists can pin down
the causal effects. So in 2001 all of a
sudden Norwegian tax records became easily
accessible online which meant everyone
in the country could see what everyone
else was earning. According to the income
comparisons model this change in transparency
can widen the gap in well-being between
rich and poor individuals they use a bunch of
survey data and they show that the high
transparency so knowing other people's incomes
actually increase the gap in happiness
between rich and poor individuals by 29% and
increase the gap in life satisfaction by 21% so
incomes didn't change the only thing that
changed was you knew other people's incomes
in your own country so if money was the only
thing that mattered for happiness more money
is more happiness then these things shouldn't
really matter but what's going on here
is we just don't care about the absolute value
of money what we care about in terms of
happiness and satisfaction is our comparison to
others like you might score a 95% on the exam
on Monday great but what was if you find
out everyone else scored more than 95% on the
exam are you still as happy as you were
before maybe but think about it so this led
to something called the Easterlin paradox and
this the idea does more economic growth make
us happier if it is this correlation and
also a causal effect of getting more money
making people happier we should strive for policy
to increase economic growth across the world
however easel in and o'connor quoted in
one of their later versions 2022 saying at
a point in time happiness varies directly with
income both among and within nations but
over time the long-term growth rates of happiness
and income are not significantly related.
So people don't really care so much in the
long run about their growth of income. What
they care about is their relative standards
and the people they're comparing themselves
against, which is usually their own society
and people within their own country. So that's
why we find that even in these richer
countries, people are not happier than those
in poorer countries, because it's all about
this relative standard. So there's a couple
of reasons for this. So as I said
before, one possible explanation is a
reference point that you're using
to compare against. So even though
you're, you know, $100,000 richer than
you were before, everyone else is
$100,000 richer, you haven't gained in
standing at all. So that's not going to change
your happiness. Another possible
explanation is something called the hedonic
adaption model, adaptation model, the hedonic
treadmill. So people get, you know, more money,
they can buy nicer So things consume more,
but they get used to that very quickly and
go back to a baseline. So even though it
increases, happiness eventually returns
to the baseline. It's a short spike
in happiness. So I've experienced
this recently, not in terms of
the adaptation, but my income from
being a grad student to a professor
jumped dramatically. And my expectations
of the things I was able to consume
changed a lot. Whether my happiness
went back to the baseline or
not is another question entirely,
but this is something that a lot of
people report. There's a number of
criticisms of this model. The first is
happiness surveys. What the hell does
it mean to ask someone how happy
you are from a scale of 1 to 10?
What does a 7 mean? I don't know. People give different
answers in the morning to the night,
depending on if they're about to
wake up or go to bed. These are just
highly variable, self-reported answers. Some people say
they should be more trusted than others.
I'm sceptical. There's a couple of
economists up at the University of Michigan,
Betsy Stevenson and Justin Wolfers I think
the data's been woefully misinterpreted by
Easterlin and co-authors and finally just something
to think about is Easterlin wrote the
original paper in the 90s, so in the 80s
a long, long time ago, different tech
and everything but now with access to internet
and social media, you're not just seeing
the lifestyles of people in your own community
you can see everyone's lives from around
the world so what becomes the point of
comparison if someone in a developing country
are they comparing themselves to their
fellow you know countrymen or they going on you
know tick tock and seeing what a bunch of
you know americans and and i shouldn't classify
the uk gdp is the same as america so a
bunch of americans and comparing themselves
to americans and what does this mean does
this mean now that it's actually better to have
more income than it did before i don't
know the answer to that i think it's a question
worth exploring for sure that's all i have
for today we're going to do the kahoot on
monday before we jump into revision for
the exam. Enjoy your weekend. Feel free to
reach out if you've got questions. I'll try
my best to access my email. Otherwise, I'll
see you all on Monday. I don't know why
I'm closing up. What's up, Eric? So, about the
class like...