>> I'll now describe a somewhat deep result in economics.
If agents have preferences
that satisfy the properties we've been talking about,
monotonicity, completeness, transitivity and continuity,
then we can represent the preferences in terms
of what's called a utility function.
There's a function, "U," with the following properties.
In case you don't remember what a function is, it's simply a way
of assigning a number, for our purposes, to objects in a set
so the objects in the set will be alternatives.
And we're going to assign a number to alternatives.
Let me give you two numbers, A1 and A2.
I'll take U of A1 and U of A2.
So U of A1 is a number that's going to be assigned
to the alternative A1.
We'll say that's the utility of A1.
And similarly, for A2.
Now, I'm going to show you
that we may assign a bigger number to A1 than A2.
And, in fact, under these properties,
the utility function will assign a bigger number to A1 than A2.
Weakly bigger.
That is, greater than or equal.
If, and only if, A1 is weakly preferred to A2.
So let's go through that.
I know it's somewhat detailed.
Let's talk about the components individually.
First off, we have a preference relation.
It's the fundamental relation or the fundamental element
of describing household behavior.
We know households have preferences
that by assumption satisfies certain properties.
Given those properties, we can assign numbers to alternatives.
For a particular household, the number I'll assign
to one alternative will be at least as big
as the number I assigned to a different alternative if,
and only if, the household prefers the first alternative.
This is quite useful in classical economics
and economics more generally,
including economics with frictions.
Because it allows us to take the somewhat abstract notion
of preference relations and map them into numbers.
We'll call "U" the utility function.
And we'll call the number the utility.
So if A1 is at least as good as A2 according
to the preference relation, then the utility of A1 is
at least as big as A2.
This is a way of describing a household's preferences in terms
of a simple function that gives us numbers to alternatives.
Let me mention two caveats.
One, you may find this unrealistic
because you may not know your own utility function.
You may not be able to easily assign a number
to the alternative where it's raining, the wrong party's
in power and the Packers lost.
However, as long as you have preferences over alternatives,
as an economist, I know there is a utility function representing
your preferences.
Instead of me saying, you prefer A1 to A2, I'll say the utility
of A1 is at least as high as the utility of A2.
We'll see how that works later in detail.
Second caveat, there does not exist a unique utility function
representing a given set of preferences.
If some function assigns a higher number to one alternative
than to a preferred alternative, there's many such functions
which do equally well.
I can add two to each side.
I can multiply by seven.
I can do various things.
So there does not exist a unique way
of assigning numbers to alternatives.
But there exists at least one way of doing it, which is useful
because it assigns a higher number
to preferred alternatives.
And we're going to use that later.
So now we have this cost of the utility function,
which will prove very useful in applications.
Both to the classical theory of markets and to markets
with frictions that we'll talk about later.
Before we put it to work in applications, however,
let me describe a couple of other properties.
So far we have properties which we need in order
to generate the costs
of the utility function in the first place.
When I say we need them, these properties are sufficient.
So we have this utility function.
I want to describe a notion called an indifference curve.
The way to do that is
to restrict attention to commodity space.
By which I mean, sets of alternatives
that describe bundles of goods, X1 and X2.
It applies to a bundle X1, X2, X3, X4, et cetera.
However, to draw it on the board,
let's restrict attention to R2.
That is to say, two dimensions.
For example, X1 could be beer.
X2 could be pizza.
And any point in commodity space --
say, X1 bar, X2 bar -- is a possible bundle.
A possible commodity bundle that you might consume.
You have preferences over these bundles.
Some other point -- say, more beer, but less pizza --
we'll use X1 double bar and X2 double bar
to represent that alternative.
Gives you some amount of utility as well.
It may be more.
It may be less.
It may be the same.
If we consider bundles X1 bar,
X2 bar that give the same utility as each other,
as described by this equality,
we know the household is indifferent
between those two bundles.
That's what it means to say they generate the same utility.
The set of all such alternatives generating the same utility is
called an indifference set.
It could look in general in some particular way in this diagram.
But if we assume a concept called convexity,
the indifference curve must look like that.
Whereby "that," I mean it's convex to the origin.
So although preferences are quite general,
this regularity condition called convexity is useful
for reasons that we'll see later.
The concept of convexity -- let me define it precisely.
Take any two points on an indifference curve --
such as the two illustrated in this diagram --
showing them up on a line.
And take a point in between them.
That point is an average, could be a weighted average,
of the two original points.
We say preferences are convex when the utility
from the average exceeds the utility from the extremes.
I'll describe that in a little bit more detail.
First, it's important
to understand there is a whole family of indifference curves.
Through any point, there will be an indifference curve.
As we move in this diagram towards the northeast,
agents are getting more of both.
That is, the household's getting more beer and more pizza.
According to our monotonicity maintained hypothesis,
that means they're better off.
So the utility assigned to bundles
to the northeast will be greater than the utility assigned
to bundles to the southwest.
When I say preferences are convex,
I mean that the point that's a weighted average
of two other points on the same indifference curve will give
more utility or will lie on a higher indifference curve
than the original two extreme points.
That's useful for reasons that we'll see later.
But it basically describes situations where it's easier
to predict household behavior.
Okay. We've talked about preferences.
Now we're going to talk about constraints.
One of the nice things about economics, it allows us
to separate preferences from constraints.
Of course, the interaction of preferences
and constraints is what determines behavior.
Let's move on now to talk about constraints,
then we'll bring it altogether.
If we restrict attention to commodity bundles --
that is, vectors X1, X2 --
that are two dimensional so that I could draw it easily
on the board.
I'm now going to describe constraints in terms of budgets.
There's two ways to do this.
I'll start with the more simple approach first.
I'm going to say there's prices.
P1 for good X1.
And P2 for good X2.
Your total expenditure on the two goods is P1X1 plus P2X2.
So, for example, if pizza cost $1 a slice
and you have 10 slices, you've spent $10 on pizza.
I could add up the price times the quantity
of each good you consume or each good you have in your bundle,
I'll get the total expenditure.
For now I want to write the income allocated to be spent
on these two goods as uppercase "I."
Basically, you come into the problem,
it's already been determined you're going to spend "I"
on the two commodities.
It's a question of allocating
that expenditure between the two.
And what we've written here is that your total expenditure
on the two goods is equal to the amount
of income that's been allocated
to expenditure on those two goods.
What's to be determined is how much of each
of the two goods will be purchased or will be consumed.
Okay. Let me represent this budget equation
in the plane as follows.
First, isolate X2 on one side of the equality.
So X2 is going to be the income divided
by P2 minus the price ratio of P1 over P2 times X1.
If you look at that --
and remember your high school graphing techniques --
you'll see that X2 is a linear function of X1 with intercept I
over P2 and a slope minus P1 over P2.
So the budget equation is described by a line like this.
That's called the budget line.
Now, I'm only considering the positive quadrant
because I don't want to consider negative quantities
of beer or pizza.
Therefore, in this positive quadrant,
the set that I've now shaded in describes all the bundles
which do not violate your budget.
Since we've already assumed you're going
to allocate a certain amount of income, "I," to these two goods,
in fact, we know you're going
to choose a point on the budget line.
That's all there is to it.
This line describes your constraints in terms
of what bundles are affordable.
Let me describe a slightly different way
of drawing the budget line.
What I want to assume now is the following.
You don't have some income that you've allocated
to expenditure on these two goods.
Rather what you have is an endowment.
This will be very useful later when we talk
about labor supply and credit markets.
So suppose the household has some endowment X1 bar X2 bar.
That's a point in commodity space.
Just like we saw earlier, I start the consumer
out with a point in commodity space.
And now what I want to say, the consumer is allowed
to choose different points in commodity space subjected
to a budget constraint.
So let me describe it as follows.
What is the value of your endowment?
Well, it's P1 times X1 bar plus P2 times X2 bar.
That's a way to describe your income in the previous picture.
"I," the income that could be allocated to these two goods,
in fact, could be thought
of as being the value of your endowment.
What I want to say is the consumer can choose different
points as long as P1X1 plus P2X2 equals the value
of your endowment.
So this looks a lot like the previous description
of a budget line.
I've simply gone into a little bit more detail
about where it is that the, I, income comes from.
Here it's the value of your endowment.
Let's do what we did before and isolate X2
on the left-hand side of this equation.
So X2 is, first off, the value
of your endowment divided by the price P2.
Subtract P1 over P2 times X1.
So, in fact, it's exactly what we had before.
The only difference being that the intercept now is written
out explicitly in terms of the value of your initial endowment.
If I draw the budget line, it looks the same as before.
And, hence, given the endowment X1 bar X2 bar,
you're allowed to choose any point.
That is, you can afford
to choose any point on your budget line.
The nice thing about writing it this way is we observe
immediately that all that matters is the price ratio.
You could see, for example, that P1 over P2 appears here.
The P2 is cancelled there.
And P1 over P2 appears there.
So a big result.
All that matters is the ratio or the relative price
as we call it in economics.
In particular, suppose I doubled prices of good 1 and good 2.
You can see that would have no effect on your budget line.
The value of your endowment will be twice as big.
But any point you want
to consume would cost twice as much.
It's not going to affect your affordable alternatives.
That's an important concept.