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Budget Constraints & the Marginal Rate of Transformation (derived from video lecture by J. Gruber) - YouTube
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Today we're going to continue
our discussion of consumer choice.
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If you remember the set-up from last time,
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the main motivation is you're trying
to understand what underlies demand curves,
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how consumers ultimately decide
to trade off price and quantity of goods.
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We said that ultimately that came from
the principle of utility maximization,
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and that utility is maximized
when individuals maximize the utility function,
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which is this mathematical
representation of preferences.
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And last time we talked about
how if individuals were unconstrained
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how they choose what they want,
they would just like more of everything,
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and their ranking across different bundles
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would depend on that underlying utility function.
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Now, of course, what's stopping individuals
from consuming everything they want
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is their budget constraints.
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And so today we're going to turn
to the second part of the problem,
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which is talking about budget constraints.
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Now, we're going to make a very simplifying
assumption here for most of the semester,
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which is we are going to assume
that your income equals your budget.
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That is, you spend your entire income.
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That is, we're going to ignore
the possibility of savings.
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So let's say that your parents, probably
a good model is you guys,
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you guys probably aren't in saving mode.
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You've got some budget saved
from your parents. Let's call it y.
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And let's say that your parents give you some
budget at the start of the semester, y,
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and they say this is your money you have to spend,
say each month or for the whole semester.
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And let's imagine that you have to allocate
that budget only across two goods,
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pizza and movies.
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So once again, unrealistic, but this is
the kind of simplifying assumption
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that lets us understand
how people make decisions.
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So that gives you your budget constraint.
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You've got some income y
that your parents have given you,
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and you can allocate that
across pizza and movies.
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So how do you allocate that?
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Well, you can buy movies,
the number of movies you can get,
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plus the number of pizzas.
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Well, how many of each you can get,
that depends on their price.
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In particular, budget constraint is the number
of movies times the price per movie
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plus the number of pizzas
times the price for pizza.
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That's your budget constraint.
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It's the number of movies
times the price per movie
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or the number of pizzas
times the price for pizza.
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And this is easiest to see graphically.
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If you go to figure 5-1, this is
a graphical illustration of a budget constraint.
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Now, let's just carefully talk through this
for a moment.
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You're going to be really good
at dealing with budget constraints.
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You're going to have to be this semester.
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So let's carefully talk
about where this comes from.
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OK, the x-axis is going to be
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how many movies you could have
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if all you did with your income
was consume movies.
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Well, if all you did with your income
was consume movies,
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you could have y over p sub m movies.
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If you decided to devote
your income solely to movies,
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then you could have y times p sub m,
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y over p sub m movies.
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If instead you decided
to devote all your income to pizza,
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then you could have y over p sub p pizzas.
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So the y-axis is going to be
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the point where you consume
zero movies and all pizza.
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It's going to be where you devote
your entire budget to pizzas.
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And then there'll be some combination
in between, which is our budget line.
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Which is the combinations of pizzas
and movies you can consume
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given your total income y.
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So basically and the slope of that line
is going to be the price ratio.
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Or the negative of the price ratio.
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The slope of that line is going to be
minus pm over pp.
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So basically it's the negative
of the price ratio,
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minus the price of movies
over the price of pizzas
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because they're in the denominators
as you said,
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because as the price goes up,
the quantity goes down.
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So the negative of the price ratio of the
price of movies to the price of pizzas is the slope.
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So let's just do a simple example.
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Imagine that income equals $96.
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Imagine your parents give you $96,
say a month or a semester.
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Imagine that the price of movies is $8
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and imagine the price of a pizza is $16.
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It's a good pizza.
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So what this means is that
with your income of $96,
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you could either get 8 pizzas or 12 movies.
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So that means that the price ratio
of the slope of your budget constraint
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is minus 1/2.
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The price ratio is minus 1/2.
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So the slope of that budget constraint
is minus 1/2.
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Now, we have a name for this slope.
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We're going to call this
the marginal rate of transformation.
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The marginal rate of transformation
is our label for this slope.
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Now, why do we use that name?
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Well, it means that's the marginal rate
at which you can transform pizzas into movies.
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The rate at which you can turn
pizzas into movies.
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Now, once again, like I talked about
last time, you're not an alchemist.
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You're not actually turning pizzas into movies.
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But the market essentially is giving you a
rate at which you can do that given a budget,
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given that you have a certain amount of money.
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Given that you have a certain
amount of money, $96,
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and given the prices that you face in the market,
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you could transform pizzas into movies
by trading one pizza for 1/2 a movie.
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Now, once again, you're not actually doing
the physical transformation,
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but that's the trade-off that you face
when you're trying to transform one to the other.
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So effectively, it's the same
as if you're trading them for each other.
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As I talked about last time.
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it's essentially the same as you're trading,
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and that's because of the key economic concept
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we'll come back to over
and over again in this course,
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the concept of opportunity cost.
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The opportunity cost is
the value of the forgone alternative.
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The value of the forgone alternative
is the opportunity cost.
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So basically what that means is
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if you decide to forgo a pizza,
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that's the same as forgoing two movies.
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Likewise, if you decide to forgo a movie,
it's the same as forgoing half a pizza.
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So the opportunity cost of a movie,
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what essentially the movie is costing you,
is 1/2 a pizza.
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Now, really it's costing you $8
and a pizza costs you $16.
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But when we think about trading off goods,
the opportunity cost of that movie
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is that you've forgone the ability to eat half a pizza.
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That's the opportunity cost of the situation.
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So that's basically how we're going
to think about this trade-off.
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We're going to think about trading off goods
as the opportunity cost
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of consuming one good instead of another.
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The opportunity cost of that movie
is that you haven't gotten to eat 1/2 a pizza.
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The opportunity cost of the pizza is that
you've forgone seeing two movies.
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And the reason is because
you have a fixedbudget.
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If you had an infinite budget,
there'd be no opportunity cost.
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But because you have a fixed budget
and you have to allocate that budget,
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there's an opportunity cost.
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If you choose not to decide,
you've still made a choice.
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I don't know whether that quote's
due to Shakespeareor Rush, I'm not sure.
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I have to look that up.
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But basically, if you choose to have one thing,
then by definition you're forgoing another.
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