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## Utility maximization with indifference curves

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# Budget line

## Video transcript

What I want to do in this video is introduce you to the
idea of a budget line. Actually, probably isn't a new idea. It's a derivative idea of what you've seen and often in an
introductory algebra course where A, you've gotten a
certain amount of money and you can spend it on a
certain combination of goods. What are all the different possibilities that you can actually buy? That's really what a budget line is. Let's say that you have an income and I'll do it both in the abstract and the concrete. I'll do it variables and then I'll also do
it with actual numbers. Lets say your income, your
income in a month is Y and lets say that you
spend all of your money. Your income is equal to your expenditures. Assuming in our little model here that you're not going
to be saving any money. To show how overly simplified
we can make a model we are going to only assume that you can spend on two different goods and that's so that we can actually plot all the combinations on
a two dimensional surface like the screen over here. Obviously, most people buy many more or they at least are
choosing between many, many more than two goods. But let's say you can
choose between 2 goods and let's just take goods
that we've been doing using in recent videos. That 2 goods that you buy are
either chocolate or fruit. You could buy chocolate by the bar or fruit by the pound. What are going to be your expenditures assuming you spend it all
on chocolate and fruit? Well, there's going to be the amount that you spend on chocolate will be the price of chocolate times the quantity of chocolate you buy which is the number of bars. And then the amount you spent on fruit will be the price of fruit per pound times the quantity of fruit. For example, if Y = $20 a month and the price, actually
we'll plot this in a second, the price of chocolate
is equal to $1 per bar and the price of fruit
is equal to $2 per pound. I think these were the prices I used in a per pound of fruit. Then all of a sudden, you
would know what this is, you would know what this is and this is. You know what the Ps are and the Y and then you could
actually graph one of these quantities relative to the other. What we can do is, and let's do that, we can graph the quantity
of 1 relative to the other. Why don't we put the quantity of chocolate on this axis over here and
let's put the quantity of fruit on this axis over here. First, if we wanted to
graph it I like to put it, since I've put quantity of chocolate on the vertical axis here, I'd like to solve this equation
for quantity of chocolate as a function of quantity fruit and it should make it pretty
straight forward to graph. Let's try that out. First, I'm just going to rewrite this without expenditures in between. We have our income, our income Y = price of chocolate times the quantity of chocolate plus the price of fruit
times the quantity of fruit. Now, I want to solve for
the quantity of chocolate. Let me make that orange so we know that this is
this one right over here. If I want to solve for that, the best way I could isolate
it one side of this equation. Let me get rid of this this
yellow part right over here and the best way to do that is to subtract it from both sides. Let's subtract the price of fruit times the quantity of fruit and I could substitute
the numbers in first and that might actually
make it a little bit easier to understand but I like to keep it general first. You see, you don't have to
just use with these numbers you could just see the
general result here. I'm going to subtract it
from the left hand side and the right hand side and the whole point is to get rid of it from the right hand side. This cancels out, the left hand side becomes your income
minus the price of fruit times the quantity of fruit. This is going to be equal
to your right hand side which is just the price of chocolate times the quantity of chocolate. Now if we want to solve for
the quantity of chocolate we just divide both sides
by the price of chocolate and then you get it,
and I'll flip the sides. You get the quantity of chocolates, is going to be equal to your income, your income divided by
the price of chocolate minus the price of fruit
times the quantity of fruit all of that over the price of chocolate. All over that over the price of chocolate. We can actually substitute
these numbers in here and then we can actually
plot what essentially this budget line will look like. In our situation, 20, Y = 20, the price of chocolate is equal to 1. Price of chocolate is equal to 1. This term right over here, $20 per month divided by $1 per bar which would actually give
you 20 bars per month if you work out the units. This term right over here
just simplifies to 20. This is actually an interesting term, your income, your income in
dollars divided by the price of an actual good or service. You could view this term right over here as your real income. The reason why it's called real income is it's actually pegging
what your earnings to what you can buy. It's pegging it to a certain real goods, it's not tied to some
abstract quantity like money which always has a changing buying power. What you could buy for $20
in 2010 is very different than what you could buy for $20 in 1940. Here, when you divide your income, divide it a by a price of some good it's really telling you your
income in terms of that good. You could view your
income as $20 per month or you could view your income if you wanted your
income in chocolate bars. You could say my income is, I could buy 20 chocolate bars each month. So I could say, my income
20 chocolate bars per month. They would be equivalent to you assuming that you could
sell the chocolate bars for the same price you could buy it and that's somewhat of an assumption. But you could say I have
the equivalent income of 20 bars a month. You could have also done it in fruit. I have the equivalent
income of 20 divided by 2, 10 pounds of fruit a month. It's trying your income to real things, not the abstract quantity like money. Anyway, this is going to be equal to, let me write it over here. My quantity of chocolate is going to be equal to this
term right over here as 20. If you wanted to do the units, it would be 20 bars per month and you could do a little
bit of dimensional analysis to come up with that. You could treat the
units just like numbers and see how the cancel out. 20 bars per month minus the price of fruit divided by the price of chocolate. $2 per pound of fruit. The price of fruit is going to be $2 and I actually want to look at the units because that's interesting. Let me write it here. The price of fruit is
equal to $2 per pound. Let me write it this way. $2 per pound of fruit, I'll show you how the units cancel out. Then we're dividing that
by the price of chocolate. Dividing it by the price of chocolate which is equal to $1 per bar of chocolate. Now, obviously the math is
fairly straight forward. We just get 2, but the units
are a little bit interesting. You have a dollar and the
numerator of the numerator and a dollar, the numerator
of the denominator, those will cancel out. You could actually view this as, this is going to be the same thing just to look at the units. This is going to be, this is the same thing as the
numerator times the inverse times the reciprocal of the
denominator right over here. You could say $2 per pound times, the reciprocal of 1 is just 1, times 1 bar per dollar. Then the dollars cancel out and you are left with 2
bars per pound of fruit. What we've actually done over here, this term right over here, it gives us bars of
chocolate per pound of fruit. It simplifies to 2 bars of
chocolate per pound of fruit. It's actually giving
you the opportunity cost of a pound of fruit. It's saying hey, you
could buy a pound of fruit but you'd be giving up
2 bars of chocolate. Because the price, you could
get 2 bars of chocolate for every pound of fruit. You could view this as the relative price, this right over here is the relative price of fruit in this example. It's telling you the opportunity cost, it's telling you how much fruit cost in terms of chocolate bars. Regardless, that number is
fairly straight forward, it was just a 2. Minus 2 times the quantity of fruit. This is fairly straight forward to plot. If the quantity of fruit it 0, our quantity of chocolate is 20. This is going to be 20 over here. This is 20 and this is going to be 10. This is 15, this is 5. This is a point on our
budget line right over there. There is multiple ways that
you could think about this. One way you could say is
if you buy no chocolate, if the quantity of chocolate is 0, what is going to be the quantity of fruit? Then you could solve this
or you could just say, "Look, if I have $20 a month "then I'm going to spend it all on fruit. "I can buy 10 pounds of fruit." So to say that this right over here is 10. Let's say this right over
here is 10, this is 5, so this is also on our budget line and every point in between is going to be on our budget line. Every point in between is
going to be on our budget line. Another way you could have done this and this comes straight
out of kind of your typical algebra 1 course. You could say, in this case, if you view this as the Y axis, you say your Y interceptor, you say, "My chocolate
quantity interceptor is 20 "and then my slop is negative 2. "My slope is negative 2." For every extra pound of fruit I buy I have to give up 2 pounds of chocolate. You could also view this as
the opportunity cost of fruit. You see this slope as we go forward, if we buy one more pound quantity of fruit we're giving up 2 bars of chocolate. One statement I did just make, I said every point on
this line is a possibility and I can only say that if we assume that both of these goods
are divisible goods which means we can buy
arbitrarily small amounts of it, that we could buy 10th
of a bar of chocolate on average especially. Or we could buy 100th of a pound of fruit. If they weren't divisible,
they're indivisible then only the whole quantities would be the possibility points. We'll just assume they're divisible, especially even if the store only sells indivisible bars of chocolate. If you buy one bar of
chocolate every 4 months, on average you're buying .25
bars of chocolate per month. Even that, on average, almost anything, almost anything here is divisible. This line right over here shows all of the combinations we can buy. All of the combinations of the divisible goods we could buy if we spend all of our money. That right over there is our budget line. That is our budget line. That is our budget line. And any combination out
here is unaffordable. We don't have enough money for that. Any combination down here is affordable. Actually, we would end up with extra money if we're below the budget line. This isn't all that
different than what we saw with the production
possibilities frontier. Remember, we had a curve
that really showed all of the if we were producing 2 goods, what combinations of
goods we could produce. Anything on that curve for the productions possibility
frontier was efficient. Anything outside of it was unattainable and anything inside was
attainable but inefficient.