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PROFESSOR: All right, so today we're going to continue our
discussion of consumer choice.
If you remember the set-up from last time, the main
motivation is you're trying to understand what underlies
demand curves, how consumers ultimately decide to trade off
price and quantity of goods.
We said that ultimately that came from the principle of
utility maximization, and that utility is maximized when
individuals maximize the utility function, which is
this mathematical representation of preferences.
And last time we talked about how if individuals were
unconstrained how they choose what they want, they would
just like more of everything, and their ranking across
different bundles would depend on that
underlying utility function.
Now, of course, what's stopping individuals from
consuming everything they want is their budget constraints.
And so today we're going to turn to the second part of the
problem, which is talking about budget constraints.
Now, we're going to make a very simplifying assumption
here for most of the semester, which is we are going to
assume that your income equals your budget.
That is, you spend your entire income.
That is, we're going to ignore the possibility of savings
until about the third lecture from the end.
Now, this turns out not to be a terrible assumption for the
typical American.
The typical American doesn't save. So actually, it's not a
terrible assumption for us to work with if we think about
typical consumers.
In practice, savings is going to turn out to be a very
critical part of what we're going to do to think about
economics, so we'll come back to that.
But we're going to ignore savings for now and assume
that your budget equals your income.
So let's say that your parents, probably a good model
is you guys, you guys probably aren't in saving mode.
You've got some budget saved from your parents.
Let's call it y.
And let's say that your parents give you some budget
at the start of the semester, y, and they say this is your
money you have to spend, say each month or
for the whole semester.
And let's imagine that you have to allocate that budget
only across two goods, pizza and movies.
So once again, unrealistic, but this is the kind of
simplifying assumption that lets us understand how people
make decisions.
So that gives you your budget constraint.
You've got some income y that your parents have given you,
and you can allocate that across pizza and movies.
So how do you allocate that?
Well, you can buy movies, the number of movies you can get,
plus the number of pizzas.
Well, how many of each you can get, that
depends on their price.
In particular, budget constraint is the number of
movies times the price per movie plus the
number of pizzas --
plus the number of pizzas times the price for pizza.
That's your budget constraint.
It's the number of movies times the price per movie or
the number of pizzas times the price for pizza.
And this is easiest to see graphically.
If you go to figure 5-1, this is a graphical illustration of
a budget constraint.
Now, let's just carefully talk through this for a moment.
You're going to be really good at dealing with budget
constraints.
You're going to have to be this semester.
So let's carefully talk about where this comes from.
OK, the x-axis is going to be how many movies you could have
if all you did with your income was consume movies.
Well, if all you did with your income was consume movies, you
could have y over p sub m movies.
If you decided to devote your income solely to movies, then
you could have y times y over p sub m movies.
If instead you decided to devote all your income to
pizza, then you could have y over p sub p pizza.
So the y-axis is going to be the point where you consume
zero movies and all pizza.
It's going to be where you devote your
entire budget to pizzas.
And then there'll be some combination in between, which
is our budget line.
Which is the combinations of pizzas and movies you can
consume given your total income y.
And the slope of that line is going to be the price ratio.
Or the negative of the price ratio.
The slope of that line is going to be minus pp over pm.
The slope of that line is going to be the change in the
ratio of the price of pizza to the price of movies.
OK, what am I doing wrong here?
Negative of the price ratio.
Have I got this right?
Rise over run.
Yeah.
Have I got this right?
Yeah.
AUDIENCE: I think that the pp and the pm's are in the
denominators, because it's y over pp [INAUDIBLE PHRASE].
PROFESSOR: Right.
It's the denominators, that's what I did wrong.
Right, so it is pm over pp.
Sorry, my bad.
OK, right.
Because they're the denominators.
Because it's the rise over run in terms of the quantity.
So that's what I did wrong.
OK, so basically it's the negative of the price ratio,
minus the price of movies over the price of pizzas because
they're in the denominators as you said, because as the price
goes up, the quantity goes down.
So the negative of the price ratio of the price of movies
to the price of pizzas is the slope.
So let's just do a simple example.
Imagine that income equals $96.
Imagine your parents give you $96, say
a month or a semester.
Imagine that the price of movies is $8 and imagine the
price of a pizza is $16.
It's a good pizza.
So what this means is that with your income of $96, you
could either get eight pizzas or 12 movies.
So that means that the price ratio of the slope of your
budget constraint is minus 1/2.
The price ratio is minus 1/2.
So the slope of that budget constraint is minus 1/2.
Now, we have a name for this slope.
We're going to call this the marginal rate of
transformation.
The marginal rate of transformation is our label
for this slope.
Now, why do we use that name?
Well, it means that's the marginal rate at which you can
transform pizzas into movies.
The rate at which you can turn pizzas into movies.
Now, once again, like I talked about last time, you're not an
alchemist. You're not actually turning pizzas into movies.
But the market essentially is giving you a rate at which you
can do that given a budget, given that you have a certain
amount of money.
Given that you have a certain amount of money, $96, and
given the prices that you face in the market, you could
transform pizzas into movies by trading one
pizza for 1/2 a movie.
Now, once again, you're not actually doing the physical
transformation, but that's the trade-off that you face when
you're trying to transform one to the other.
So effectively, it's the same as if you're trading them for
each other.
As I talked about last time.
it's essentially the same as you're trading, and that's
because of the key economic concept we'll come back to
over and over again in this course--
the concept of opportunity cost.
The opportunity cost is the value of the forgone
alternative.
The value of the forgone alternative is the opportunity
cost. So basically what that means is if you decide to
forgo a pizza, that's the same as forgoing two movies.
Likewise, if you decide to forgo a movie, it's the same
as forgoing half a pizza.
So the opportunity cost of a movie, what essentially the
movie is costing you, is 1/2 a pizza.
Now, really it's costing you $8 and a pizza costs you $16.
But when we think about trading off goods, the
opportunity cost of that movie is that you've forgone the
ability to eat half a pizza.
That's the opportunity cost of the situation.
So that's basically how we're going to
think about this trade-off.
We're going to think about trading off goods as the
opportunity cost of consuming one good instead of another.
The opportunity cost of that movie is that you haven't
gotten to eat 1/2 a pizza.
The opportunity cost of the pizza is that you've forgone
seeing two movies.
And the reason is because you have a fixed budget.
If you had an infinite budget, there'd be no opportunity
cost. But because you have a fixed budget and you have to
allocate that budget, there's an opportunity cost. If you
choose not to decide, you've still made a choice.
I don't know whether that quote's due to Shakespeare or
Rush, I'm not sure.
I have to look that up.
But basically, if you choose to have one thing, then by
definition you're forgoing another.
Now, to understand the budget constraint, let's talk about
what happens when we shock the budget constraint.
Let's imagine the price of pizzas rose from $16 to $24.
Pizzas got really expensive.
We decided we only want gourmet pizzas or something.
The price of pizzas went from $16 to $24.
What does this do?
Well, let's look at figure 5-2.
It'll show what that does.
What that does is it says our new budget constraint, instead
of being 16p plus 8m equals 96, which is what the budget
constraint was in our example, it's now 24p
plus 8m equals 96.
That's the new equation for the budget constraint.
Or, more relevantly, the slope of the budget constraint has
flattened from minus 1/2 to minus 1/3.
The slope has fallen from minus 1/2 to minus 1/3.
The price ratio has been reduced from 1/2 to 1/3.
Now, forget utility for a second.
Forget the fact that we thought about utility.
Just looking at this, can you tell whether you are better or
worse off from this price change?
You shook your head no, why not?
AUDIENCE: Because you don't know if you like pizzas enough
to get any.
PROFESSOR: OK, well in particular,
you can almost tell.
Who's the only person who doesn't care
about this price change?
AUDIENCE: [INAUDIBLE]
PROFESSOR: No, no no.
Think about consumers, people with different preferences for
pizzas and movies.
What would your preference for pizzas and movies have to be
for you not to care about this price change?
All movies.
So long as you care about pizza at all,
you're worse off.
So in fact, the answer is your opportunity set has been
restricted.
So we can think about the opportunity set.
Your opportunity set is the set of choices you can make
given your budget.
Before, you could make choices all the way up
to the upper line.
Now your set of choices that are
available have just fallen.
Now, you're no poorer-- it's not like your parents
have cut you off.
They still give you the $96.
But you're effectively poorer.
You're effectively worse off, and why is that?
Because the set of things you could afford with that $96 has
just been restricted.
And unless you truly have no value on pizza, unless all you
care about's the movie--
you're gluten and cheese allergic or something, you
have no value on pizza--
then you're worse off.
Your opportunity set's restricted.
And that's the key insight here, is that you are worse
off because the price has increased.
A price increase makes you worse off.
It restricts your opportunity set, because with the same
amount of income, you can now afford fewer goods.
Your opportunity set has been restricted.
Likewise, now let's talk about what happens
when your income falls.
That's the next figure.
Now, let's suppose your parents are pissed at you and
they cut you down to $80.
Because you didn't do something.
You don't write enough or call enough, so they
cut you down to $80.
Well, here the slope of the budget
constraint has not changed.
Because what determines the slope of the budget
constraint?
It's prices, and no prices have changed.
The slope of the budget constraint is unchanged
because prices haven't changed, but your opportunity
set is once again restricted because you
now have lower income.
So you can now afford fewer pizzas and movies.
So now, instead of being able to afford up to six pizzas and
up to 12 movies, you can now only afford up to five pizzas
and up to 10 movies because your income has fallen.
So once again, you're unambiguously worse off.
Your opportunity set has contracted.
So your opportunity set will contract whenever income falls
or whenever price increases.
And how it affects the graph will depend on whether it
affects prices, which affects the slope, or just income,
which affects the intercepts.
Now, questions about the budget constraints and
opportunity sets?
Armed with that--
Yeah, I'm sorry, go ahead.
AUDIENCE: Is the area under the curve at all
indicative of utility?
PROFESSOR:No, it's not, because that's going to be
determined by your preferences.
It's indicative of, if you will, potential utility,
because that's your opportunity set.
But as the example here points out, if you don't like pizzas
at all, you'll feel very differently than if you like
pizzas a lot.
So it's indicative of sort of your potential utility, but
not your actual well-being.
But now that's a great segue to the next step.
Let's put that together and talk about constrained choice.
Which is now, let's put together--
we know what your preferences are, we've mathematically
represented those by utility function.
We know what your budget set is, we've mathematically
represented that based on your income and prices.
Now let's put them together and talk about
how you make choices.
And the basic question you want to ask is, what's the
highest utility you can achieve given the constraints
your budget constraints put on you?
Or, graphically--
so let's say you wanted to understand this intuitively,
graphically, and mathematically.
Intuitively the idea is quite simple, I think, which is
just, what's the most you can have given the constraints
that are placed on you?
Graphically, we represent that as asking, what is the
furthest out indifference curve you can achieve?
Because remember, more is better.
Indifference curves that are further out make you happier.
So what's the furthest out indifference curve that you
can achieve given your budget constraint?
So to do that, let's actually do an example.
Let's imagine, as last time, your utility is the square
root of pizza times movies.
Once again, this has no fundamental meaning, it's just
a mathematical representation of your preferences.
So your preferences are mathematically represented by
utility equals square root of pizza times movies.
And let's have the same budget constraint
that we have up here.
Income is $96, price of movies is $8, price of pizza is $16.
Now let's go to the next graph.
Figure 5-4.
What this does is put together our indifference curve
analysis with our budget constraint analysis.
It's a little complicated.
The budget constraint line is the vertical line running from
a y-intercept of six to an x-intercept of 12.
The-- not the vertical line, the straight line running from
a y-intercept of six to an x-intercept of 12.
That's your budget constraint.
We saw that before.
Then we have here a series of indifference curves.
These curves are drawn-- these are a mathematical
representative of this utility function.
These are points among which you're indifferent if you have
that utility function.
And what we see is that the best you can do is to choose
point D. Point D, with six movies and three pizzas--
OK, that should be p on the y-axis, not C. It should be
movies on the x-axis and pizzas on the y-axis.
That should be p on the y-axis.
The best you can do is to choose a point D.
Now, to see that.
And that gives utility.
What's the value of your utility at point D?
We understand value's meaningless, but just so we
can compare, what's the value of your utility at point D?
The square root of 18.
The value of utility is the square root of 6 times 3,
which is the square root of 18.
Which is going to be square root of two, or three times
square root of two, but we'll just call it square root of 18
for comparison.
Now, let's talk about why that's the best point for you.
Let's think about some alternative points.
For instance, why is that better than point E?
Somebody raise their hand and tell me, why is point D better
than point E?
Yeah?
AUDIENCE: E is unattainable with your budget.
PROFESSOR: E would be better, that'd be great.
We'd love eight movies and four pizzas, but
we can't reach it.
So E's unattainable.
Why is it better than point A?
Point A you can afford.
So why's point E better than point A?
You could afford point A just like you can
afford point D. Yeah?
AUDIENCE: Because the utility's only root 10.
PROFESSOR: Because what?
AUDIENCE: It's only root 10.
PROFESSOR: Yes, exactly, because the utilitity's only
the square root of 10.
You're on a lower indifference curve at point A. So it's true
you could afford point A, but you're on a lower
indifference curve.
Your utility's a lower value, it's only square root of 10.
So point A is dominated by point D. What about point C?
Well, point C you have the same--
point C is just an inward shift from point D, but here
that's a dominated choice.
Once again your utility's lower.
It's the square root of 4.5 times 2.2.
And basically that's dominated because you could afford more.
So basically, the point is that the point which will make
you happiest is the point at which your indifference curve
is tangent to the budget constraint.
Because that is the point of the farthest out indifference
curve that you can reach given your budget constraint.
The tangency of the indifference curve and the
budget constraint is the point which makes you best off given
your available budget and the available prices.
And that's the point where the slope of the indifference
curve equals the slope of the budget constraint.
The tangency is the point where the slope of the
indifference curve equals the slope of the budget
constraint.
Or, more relevantly--
OK, let me stop there.
That's the graphic intuition.
With a sloping indifference curve because the slope of the
budget constraint is the optimum, because by
definition, that is the point of the furthest out
indifference curve you can reach given your budget.
The point of tangency is the point of equal slopes.
Are there questions about the graphical analysis here?
This is very, very important, so.
Yeah?
AUDIENCE: This is sort of about the graphical analysis,
but if it only matters in terms of the ordinal values of
the utility function and p and M are always positive, does it
matter if you got rid of the square root?
Would anything change if it was u equals pm?
Because the the marginal rate of substitution,
transformation, all that would still be the same, right?
PROFESSOR: In this particular example, it would not.
So actually, you're asking a great question.
Because it's ordinal, you can typically do transformations
for the ranking of bundles.
You will always get the same ranking with a monotone
transformation of the utility function.
That's exactly right.
Later in the course, we'll show different ways why the
functional form matters, and I'll show you why I did square
root, because it's going to turn out that
that's going to matter.
But for other things-- but for the ranking
bundles, you're right.
The ranking of bundles is consistent with the
transformation.
So that's the graphical.
Now let's come to the mathematical
derivation of this.
So let's talk about the mathematics of utility
maximization.
Now what I'm going to do here is, I'm going to do this sort
of casually, as is my wont.
Friday in section, you're going to work on the
underlying calculus that lies behind the mathematics that
I'm going to present here.
Now, let's talk about what it means that
these slopes are equal.
Well, remember, what is-- does anyone remember what the slope
of the indifference curve is?
What do we call the slope of the indifference curve?
Yeah?
AUDIENCE: The MRS.
PROFESSOR: The MRS. The slope of the indifference curve is
the marginal rate of substitution.
Which is defined as what?
What is the marginal rate of substitution?
The ratio of what?
AUDIENCE: [INAUDIBLE]
PROFESSOR: No, the marginal rate of substitution.
This is just about preferences.
What's the marginal rate of substitution defined as?
It's the ratio of what to what?
Yeah?
AUDIENCE: The amount of one good you have to get to give
up a unit of the other good.
PROFESSOR: I'm sorry?
AUDIENCE: The amount of one good you have to get to give
up a unit of the other good.
PROFESSOR: So graphically that's what
it's defined as, exactly.
It's the slope of the indifference curve.
Mathematically, what was it?
What was it in terms of utility?
Does anyone remember?
Yeah.
AUDIENCE: The ratio of the partials.
PROFESSOR: Ratio of the marginal utilities.
In particular, it's the negative of the marginal
utility of movies over the marginal utility of pizza.
Remember, it's the negative of the marginal utility of the
x-axis over the marginal utility of the y-axis.
So marginal rate of substitution is the rate at
which you're willing to substitute between movies and
pizza, which is a function of your marginal utilities.
If your marginal utility for movies is very high, then you
need a lot of pizzas.
Then you wouldn't trade a movie unless you get a lot of
pizza for it.
If your marginal utility of movies is very low, you'd be
happy to give up a movie even for a small
fraction of a pizza.
So that's the marginal rate of substitution.
That's about preferences only.
At the same time, we're saying that that marginal rate of
substitution is equal to--
this slope is equal to the slope of the budget
constraint.
Well, the slope of the budget constraint we call the
marginal rate of transformation, which is the
price ratio.
The slope of the budget constraint is the negative of
the price ratio.
That's where you were sort of one step ahead of us here.
So preferences give us this, the marginal rate of
substitution.
The mechanics of the market give us the marginal rate of
transformation.
And utility maximization gives us that those are equal,
because they're equal at that tangency.
At that tangency is where you get to the highest possible
indifference curve.
So at the optimum, you get that the ratio of marginal
utility equals the ratio of prices.
Now, I want to try to see you understand this a bit more
intuitively, given this mathematics.
The way I like to think about this is, think about the ratio
of the marginal utilities as the marginal benefit.
So it's the benefit of another movie in terms of pizza.
The marginal rate of substitution is the benefit of
another movie in terms of pizza.
It's how much you like that next movie relative to how
much you like that next pizza.
The marginal rate of transformation, the cost, is
the price of that next movie relative to the price of that
next pizza.
So what we're saying here is we're setting benefits equal
to the costs.
In particular, we're setting marginal benefits equal to
marginal cost.
The marginal benefit, the benefit to that next movie in
terms of pizza, has got to be equal to the marginal cost,
the cost to you in terms of that next
movie in terms of pizza.
And this notion that the optimum will be where marginal
benefit equals marginal cost will pervade
through the whole course.
When we do firm maximization, it'll be the same thing.
Any maximization we'll do in this course, any optimization,
will be about equating these margins.
Setting the marginal benefits equal to marginal costs.
Now, this is different than benefits equals cost, because
it's about the next unit.
It's saying, how do we feel about that next movie compared
to the price of that next movie.
Now, prices here we have being constant.
You could imagine prices of movies changing as you see
more, but that gets complicated.
We'll worry about that later.
For now, the price is constant.
But the marginal utilities are not constant.
Marginal utilities are obviously changing the more
moves you see.
Once again, with intuition, you have to
develop your own intuition.
The way that I like to think about this intuitively is to
actually rewrite this a little bit, and rewrite it as saying
that the marginal utility of movies over the price of
movies equals the marginal utility of pizza over the
price of pizza.
At the optimum, this'll be true.
I like this because to me this term sort of says, look, the
bang for the buck has to be the same across all goods.
For each dollar of movie expenditure, what's it buying?
What's that next dollar of movie expenditure buying you?
This is saying, what's that next dollar of pizza
expenditure buying you?
And they've got to be equal.
If the next dollar of movie expenditure buys you a lot
more happiness than the next dollar of pizza expenditure,
then you're not at the right place.
You should shift your money and spend more on movies and
less on pizza.
If the next dollar of pizza expenditure buys you a lot
more happiness than the next dollar of movie expenditure,
then you're not in the right place either.
You should see fewer movies and buy more pizza.
So basically, it's where the marginal benefit to you--
the bang for the buck of that next movie is the same as the
bang for the buck of that next pizza.
So to see this, let's go back to figure 5-4 and let's
actually think through the mathematics of a couple of
these points.
Let's take point A. At point A, you have two pizzas and
five movies.
So pizza equals two, movies equals five at point A. So
utility is equal to square root of 10 at point A.
Now, in particular, your marginal utility for pizzas at
that point--
what's the marginal utility for pizzas?
Well, we can differentiate.
That's the derivative of the utility function with respect
to prices, du/dp.
Which is going to be 0.5 times movies over the square root of
p times m, which at these values is going to be one over
the square root of 10.
That's your marginal utility of pizzas.
Your marginal utility of movies is going to be du/dm,
which is going to be 0.5 times p over square root of p times
m, which is going to be 2.5 over square root of 10.
So the marginal rate of substitution between
these two is 2.5.
The marginal rate of substitution, 2.5.
What does that mean intuitively?
Can someone tell me what that means intuitively, the
marginal rate of substitution is 2.5?
What does that mean?
Someone explain that like you'd explain it to someone
who's speaking English.
What does it mean that the marginal rate of
substitution is 2.5?
Yeah?
AUDIENCE: You'd give up one pizza for 2.5 movies.
Yes.
No, actually, the opposite.
You'd give up two and 1/2 pizzas--
the marginal rate of substitution is 2.5--
no, that's right.
You would give up 2.5--
one second, let's make sure I have this right.
Right, so you would give up one pizza to see 2 and 1/2--
no, it's the other way around.
You're getting a lot of pizza and not enough movies.
So you would give up two and 1/2 pizzas to
get one more movie.
This is confusing, OK?
You'd give up two and 1/2 pizzas to get one more movie.
That's what it means.
You would give up two and 1/2 pizzas to see one more movie,
and why is that?
Why at a point like A would you give up two and 1/2 pizzas
to see one more movie?
Yeah?
AUDIENCE: Well, if you just look at the line tangent to
the indifference curve at A, it's really
not a downward slope.
PROFESSOR: It's really steep, which means what?
AUDIENCE: Which means you get a lot more benefit--like you
don't care if you give up a lot of
pizzas to see more movies.
PROFESSOR: Exactly.
Actually, that's a great way to bring the graphics and the
intuition together.
Here's thinking of it intuitively.
I'm getting a lot of pizza at point A. I'm not
getting a lot of movies.
So I would happily give up a lot of pizza
to get my next movie.
What you pointed out is the tie to the graphics here.
The indifference curve is very steep at that point through A.
A steep indifference curve in that way means I don't really
care a whole lot at this point about how many pizzas I get,
but I care a lot about getting more movies.
So at a point like A where it's very steep, you are
willing to give up two and 1/2 pizzas to see a movie.
But what do you have to give up?
What's the market telling you you have to
give up to see a movie?
How much pizza do you actually have to give up to see a movie
in practice?
You're willing to give up two and 1/2 pizzas to see a movie,
but how many pizzas do you actually have to give up?
AUDIENCE: 1/2.
PROFESSOR: 1/2 a pizza to see a movie.
So that can't be the optimum.
You're willing to give up two and 1/2 pizzas to see a movie,
but you only have to give up 1/2 a pizza to see a movie.
So you can't be at the right place.
You should be changing your consumption bundle.
You should be changing your consumption bundle, because
the market is only asking for 1/2 a pizza to see a movie,
but you're willing to give up two and 1/2
pizzas to see a movie.
So at a point like A, you're clearly not at the optimum.
Because you are willing to make a trade.
Remember, we talked about-- we can go all the way back to the
first lecture.
The key point was-- the first or second lecture--
inefficiency happens when trades aren't
made that people value.
Here's a trade that you value.
You're willing to give up two and 1/2 pizzas to see a movie.
The market only wants 1/2 a pizza to see a movie.
You're not making a trade you value, so that's not the
efficient outcome.
Likewise, let's do the same mathematics at point B. Well,
at point B, if you do the math and work it out, you see that
the marginal utility of pizza is five over
square root of ten.
The marginal utility of movies is 0.5 over the
square root of 10.
So the MRS is 0.1.
At this point, you'd only be willing to give up 0.1 pizzas
to see a movie.
At a point like B, the indifference
curve is very flat.
You're only willing to give up 0.1 pizzas to see a movie.
But remember, the market is willing to say, look--
you can flip it around, the market's willing to say, look,
you can get a movie.
You're only willing to give up 0.1 pizzas to see a movie.
Well, that means clearly that you have too many movies and
not enough pizza.
You're clearly at that point happy to say, wow, you mean
that I can gain a whole pizza by just giving up two movies?
Heck, I'd be willing to give up 10 movies to get a pizza.
At the point I'm at right there, I'd be willing to give
up 10 movies to get a pizza.
You're telling me we only have to give up two
movies to get a pizza?
Great.
I'm going to do that trade.
I'm going to move back towards point D. So that's why this
idea of what you're willing to do, which is the MRS, and what
the market's making you do, you want to equilibrate those
to decide how much you want to consume.
Now, obviously a point like--
let's talk about point C. Point C is interesting,
because at point C, what's true?
The marginal rate of substitution is equal to the
marginal rate of transformation at point C. The
slope of the indifference curve and the budget
constraint are equal.
That's why you have to check two conditions for
optimization.
First of all, those slopes have to be equal.
Second of all, you've got to spend all your money.
So it's true there's a whole host of points-- in fact,
there's a vector running through CDE, all which are
points where the margin rate of substitution equals the
margin rate of transformation.
But only D is optimal, because you also have to
remember more is better.
You never want to leave money on the table.
So the two conditions you have to meet is that you're at the
point where your desired trade-off between pizzas and
movies is the same as the market's, and where you're
spending all your budget.
And that's the optimum.
OK, questions about that?
So that's basically how we think about optimization.
That's how we think about consumers making their
decisions deciding between consuming pizzas and movies.
Now, let's come back-- however, this is a particular
case we've looked at.
This is a case in particular where we've imposed that
there's an interior solution.
In fact, in practice you could end up in these kinds of
choices with corner solutions.
So let's take a look at the last figure, figure 5-5.
We've chosen a case where your optimal bundle includes both
pizza and movies.
But you could imagine a situation where
your optimal bundle--
and once again, that should be a p, not a c in figure 5-5,
that should be p on the y-axis--
where your optimal bundle includes
only one or the other.
So this is a particular case-- we have the same budget line
as before, which is you're trading off pizza and movies.
You have an income of 96, the price of pizzas is 16, the
price of movies is 8.
So same budget line as before.
But now your indifference curves look very different.
Now your preferences are such that you've got these linear
difference curves of the form I1, I2, I3.
You've got these linear indifference curves.
What that means is you've got--
these indifference curves mean that you have a constant rate
at which you're willing to trade off pizza for movies.
If we go back to figure 5.4, the rate at which you're
willing to trade off pizza for movies changes.
Your preferences are such that--
because the square root function as pointed out--
your preferences are such that you are willing to make
different rates of trade at different amounts.
In figure 5-5, your preferences are constant, no
matter how many movies or pizzas you have. You're always
willing to make that trade-off at the same rate.
Well, in that case you can end up with a corner solution,
where in fact, you're going to consume only six
pizzas and no movies.
And why is that?
That's because this is a person who loves pizza
relative to movies.
It's a very flat indifference curve.
They love pizza relative to movies.
And they love pizza so much relative to movies that given
the prices they face, they'll just go ahead and choose six
pizzas and no movies.
So that's a corner solution.
So mathematically, as you'll go through a section on
Friday, you're going to have to check for corner solutions.
You may solve these problems and end up with negative
quantities and be befuddled about what happens.
Well, if an answer looks wrong, it is wrong.
If you solve problems with negative quantities, that's
probably because there's a corner solution to the
problem, and actually the optimal quantity is to have
zero of one thing and spend your entire budget on
something else.
Questions about that?
So now let's think about applying this to the kinds of
decisions that you all have to make.
So I talked about this particular example of pizza
and movies.
And in fact, you might say, well, that's sort of
unrealistic.
Gee, I spend my budget on lots of things,
and how do I do this?
Well, in practice, what you'd have to do is you'd have to
draw a multi-dimensional graph and solve a
multi-dimensional problem.
And that's a bear.
But in practice, in fact, we can often think about breaking
down the choices we make into pairs of choices.
In practice, you could think about saying, look--
many people do what a lot of psychologists call mental
accounting, where basically they say, look, yes, I have a
whole budget and lots of things I can buy.
But in fact, I like to think of my budget in sort of
subcategories.
I think of a certain amount I'm willing to spend on
entertainment and a certain amount I'm
willing to spend on food.
And I take my budget and I mentally put it
in different buckets.
And within each of those buckets, you can then do the
same kind of optimization problem that we've done here.
So even though in reality we choose across a whole host of
goods, in practice what you're going to see is that people
will do this kind of mental accounting, where they sort of
divide their goods into different buckets and optimize
within each of those buckets.
What this means in practice is that in fact, if we now stop
for a second and think about the government, and how it
affects our consumption decisions, what this means is
that in practice, the government--
so, one way we typically of the government affecting
consumption decisions is through the power of taxation.
So let's say for example, the government decided
that pizzas were bad.
They caused obesity.
That there's too much obesity because people are eating too
much pizza, and we need to deal with that through a
government policy that involves taxation.
Somebody talk me through the analysis of how we'd think
about analyzing a tax on pizza given these diagrams.
Let's go to figure 5-4.
And imagine I said that we're going to place
a 50% tax on pizza.
So we're going to say that every dollar you pay on pizza,
you're going to have to pay $0.50 to the government.
Because we're really worried people are
eating too much pizza.
What would that do the budget constraint?
Yeah.
AUDIENCE: Well, effectively you're increasing
the price of pizza.
PROFESSOR: Effectively you're increasing the price of pizza
to the consumer.
AUDIENCE: The budget constraint would
shift down like this.
PROFESSOR: It's actually going to have the same effect as we
saw in figure 5-2.
In fact, I've just replicated figure 5-2.
Because in figure 5-2, the price of pizza went up by 50%,
from $16 to $24.
That's the same thing that the government's just done.
It's raised the price of pizza effectively from $16 a pizza
to $24 because instead of paying your $16, you're also
going to pay $8 to the government in tax.
So what's that going to do?
That is going to, in general, lower consumption of pizza.
So that kind of price increase is going to, in general, lower
the consumption of pizza.
So the government has tried to accomplish its goal by
shifting people away from pizza towards movies, away
from pizza toward other things.
Yeah?
AUDIENCE: Would it be meaningful to say that it also
affects the utility curves?
PROFESSOR: It would not be meaningful, actually--
it will affect your optimal choice.
In general, you will choose a different amount
of pizza and movies.
We'll talk about that next time.
But it's very important, it would not be right to say it
affects utility.
Utility is like what you're born with, it's an innate
concept about your underlying preferences.
However, you're actually getting to my point, which is
in fact, the way economists typically think about this is
the government can't affect your preferences.
But in fact, if people do mental accounting, the
government maybe can affect your preferences.
So let's say that basically the way I think about it is,
let's say I think I have a budget for food and I have a
budget for entertainment.
And let's say I think my budget for entertainment's
pretty small because I'm low income.
I've gotta have a budget for food.
And so let's say I put pizza in my budget for food, so I
allocate some of my budget for food to pizza.
And the government can then cause me to eat less
pizza by taxing it.
But what if somehow the government could get me to
think of pizza differently?
What if somehow the government could get me to think of pizza
as entertainment?
And suddenly I put it in that bucket, where I'm trading it
off not against other food, but against the fact that I
want to see a movie and I want to download stuff
from iTunes, et cetera.
Maybe then I'd buy less pizza at the same price because I'm
putting it in the bucket where I have less money.
So in other words, we've imagined a
world with two goods.
And the only way you can affect the choice across those
two goods is to lower your income or your price.
But in fact, if people have lots of different bundles of
goods, somehow I could shift you mentally from considering
pizza in a bucket where you have a lot of money to putting
pizza in a bucket where you don't have as much money.
I can lower your consumption of pizza without affecting
prices, without an obtrusive government tax policy.
This is the kind of thing that we call in
economic policy a nudge.
And there's a new book called Nudge by Richard Thaler, who's
a famous behavioral economist, basically bringing psychology
into economics.
There's this very important field now in economics called
behavioral economics, which is all about, how can we bring
the lessons of psychology into economics?
We don't do that in 14.01.
14.01's all about, we assume everybody's these perfectly
rational people who would never really be fooled into
thinking about pizza differently just because of
what the government told them.
But in fact, in reality people think about things differently
based on the kinds of information they have. And
given that tax policies can feel very intrusive-- imagine
some of you are like, wow, they're raising my pizza to
$24, that seems very intrusive.
If the government could somehow through nudging you to
think about pizza differently change your pizza consumption,
that might be a much more acceptable and palatable
policy to many people.
And that's the kind of role the government can play, or
policy-makers can play, is not just by changing your prices
or income, but by actually changing how you categorize
things mentally, they can change the choices you make.