Now let's step from indifference curves,
which is a building block
of preferences, to utility.
Now everything you need 
to know about preferences
is represented in those indifference maps.
The problem is they're pretty 
awkward to work with
when we need to actually prove theorems
and solve and understand 
how people make decisions.
That's a lot easier if we have a mathematical
representation of those preference maps.
And that's the utility function.
So the utility function is a mathematical
representation of preferences.
That's all it is.
You're going to be hearing this term 
in your nightmares for the next semester.
Utility functions.
But remember,
it's just a mathematical representation
of people's underlying preferences.
Don't be scared of it.
And the key thing is that we assume individuals
have these well-defined utility functions,
and by maximizing those utility functions
we can tell what choices they're going to make.
So for example,
suppose that I said that your
utility function over pizza and movies
was the square root of pizza times movies.
That's a utility function.
I'm going to say, 
what the hell does that mean?
Well, it doesn't mean anything, 
it's a utility function.
It's your preferences.
It's a mathematical representation 
of your preferences.
What does that mean?
What it means is 
-- it doesn't mean anything inherently,
but it tells us about your preferences.
What it tells us is that 
your preferences can be represented.
If you flip back to figure 4-1b,
it tells us those are your preferences
because you're indifferent between 
two pizzas and one movie
and one pizza and two movies.
Of course you're indifferent.
They both give a utility square root two.
But you prefer two pizzas and two movies
because that gives a utility of two.
So this is a mathematical representation
consistent with those utility indifference curves.
Not the only one.
There's other mathematical representations
that could be consistent 
with those indifference curves.
But let's posit that 
this is your utility function.
This is a mathematical representation
of your tastes.
Now what does utility mean?
Utility means nothing
in the sense that 
it is not a cardinal concept.
It's only an ordinal concept.
So if I say to you
that you have get two utils
from two pizzas and two movies,
that doesn't mean anything.
It just means that you get more 
than from one pizza and one movie.
And we can even get the ratio 
that you get square root of two more,
than you get from 
one pizza and two movies.
We can do ranking and ordinality,
but we can't assign cardinality.
I can't say how happy you are
in some abstract absolute sense 
from two pizzas and one movie.
I can't give a cardinal form preference.
But this is an ordinal ranking of preferences.
I can tell what you like 
better than what else.
That's why utility function is 
a representation of indifference maps.
They're just a mathematical tool 
for comparing bundles,
they're not some inner answer to the value 
of your soul or something like that.
Don't imbue these with too much magic.
They're just mathematical ways 
of representing preferences.
