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Risky Finance Part 2 Utility function excel example - YouTube
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- So I'm going to go ahead now
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and do a little
numerical example
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with a utility function here.
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So I've got a utility function,
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and it's a pretty simple one.
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There are lots of ways
in which economists
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model utility function.
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In this one,
the person's utility is just
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their consumption raised
to whatever number
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I've put in this cell here,
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so I'm calling this thing Theta.
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That's the name
of the coefficient.
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So when this person has zero
consumption,
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they get zero raised
to the 0.4 power.
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When they have $1,000
of consumption,
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they get $1,000 raised
to the 0.4 power
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and so on and so forth.
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And just because this is ugly
to have so many decimal places,
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let me go ahead and decrease
the number of decimal places
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to make it a little
bit less ugly.
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Okay, so you can see we
also have the concept
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of what is called
marginal utility.
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So another way
of saying people are risk averse
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is that they have decreasing
marginal utility.
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So the first $1,000
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of consumption gets this person
15.84 units of utility.
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Ah, you won't let me do that.
Okay.
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Let's go ahead--
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let's highlight all of this.
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So the first $1,000 worth
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of income gets this person 15.8
units of utility,
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and don't worry too much about
exactly what the units are.
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And the second
$1,000 gets them 5.1, the third
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$1,000 gets them 3.7. So
what I keep on doing here
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to calculate this is I take
the new value
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and subtract off
the previous value.
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And if we wanted to look
at these things graphically,
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this curve right here is how
their total utility changes
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as their income changes.
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This income--
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this curve over here is how
their marginal utility changes.
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Another way of saying
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that is the marginal utility is
the slope of that curve there.
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So this is all good
and nice and everything,
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and we can see how marginal
utility gets smaller
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and smaller and smaller
as we go up to higher
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and higher levels
of consumption.
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Two things to illustrate.
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One is I talked previously
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about we could think
about what level of consumption
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would be equivalent to the risk
that they're bearing.
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So if this person was at $5,000
worth of consumption,
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and they faced a 50/50 chance
of either being $5,000 poorer,
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which would give them
$0 consumption,
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or $5,000 richer, which would
be $10,000 consumption.
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Then what level
of utility would they get?
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Well, we would think normally
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that they would have a 50%
chance of having a low level
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of utility up here
and a 50% chance
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of having that high level
of utility there.
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So this calculation takes each
of those numbers in D4 and D14,
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multiplies by 0.5
because you have a 50% chance
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of ending up
in each of those categories
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and then figures out what level
of utility that would give you,
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and it gives you 19.9.
And you can see 19.9
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is equivalent to being somewhere
between $1,000 and $2,000.
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In particular,
we can take that 19.9
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and raise it to 1/this Theta
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and get back to the level of
income that would be consistent.
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So this person would
rather receive $1,767
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with 100% certainty than face
this gamble of, well,
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I could end up $5,000 better off
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or I could end up $5,000 worse
off and get nothing at all.
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So obviously,
they are quite anxious
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to avoid having nothing at all.
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Another way to look
at this kind of thing
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is how would people react
to different bets here.
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And so what I'm thinking
of now is a bet
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where heads they win $150,
tails they lose $50.
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So obviously, on average,
this is a winning bet.
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And in particular,
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you can see that well,
on average,
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they're going to go ahead
and win about $50.
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So this particular person,
if they face that bet--
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don't worry too much
about the exact math here--
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this person would
have higher utility
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if they have the opportunity
to take this bet.
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And you can see that kind
of relative to the level
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of income they already have,
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this prospect of like worst-case
scenario is I lose $50.
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Well, that's not too bad.
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And there is
a considerable upside.
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What if we go ahead
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and look at a case where it's
the same expected gain?
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Notice the good scenario is $100
higher than the bad scenario.
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But the swing around
is much bigger.
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Instead of -50 versus
+150 we have -500 versus +600.
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And yes, this person still
benefits from taking that bet,
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on average, but not by as much.
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An even more uncertain bet,
same expected outcome,
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the average winnings are
still the same,
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but the level of uncertainty is
much greater with Bet Number 3,
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and this person would reject
that bet because essentially,
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the downside pain
of losing $1,000 is so big
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compared to the average
expected gain of $50.
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And they're definitely
going to reject Number 4.
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So at a higher level of income,
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this person's attitudes
towards those bets changes.
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The first two bets,
they were acceptable before.
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Now, the third bet is also
acceptable for this person
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who has $15,000 worth
of consumption.
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The fourth bet is still
unacceptable to them.
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And then finally,
if we go down to someone
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who has an even higher
level of consumption,
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someone who is already
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at $35,000, they're going
to find all of these bets
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as potentially
rewarding for them.
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So notice someone's level
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of current income or current
consumption is going to affect
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how risk averse they are.
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Also, just some people have
a different psychology.
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It might be
that this Theta here,
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which is going
to tell us something
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about their risk aversion,
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is just going to be different
for different people.
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And the way I've written this,
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this Theta essentially measures
how risk tolerant someone is.
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If they have a Theta of 1,
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they're completely
risk tolerant.
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If they have a Theta of,
like,
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0.01, then they're going
to be pretty risk intolerant.
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So just people
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may have different
psychological predispositions.
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Some people are sort of very
anxious and risk avoidant.
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Other people are a little bit
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more optimistic
and risk tolerant,
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and people are just going to end
up making different decisions
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because of that in addition
to differences
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in their level of income.
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