Budget Line

published on July 18, 2020

What I want to do in this video is introduce you to the idea of a budget line and actually probably isn't a new idea it's it's kind of 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 so that's really what a budget line is so let's say that you have an income and I'll do it both in

The abstract and the concrete so I'll do it with variables and then I'll also do it with actual numbers so let's say your income your income in a month is y and let's say that you spend all of your money so your income is equal to your

Expenditures expenditures so we're assuming in our little model here that you're not going to be saving any money and 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 money more or they at least are choosing between many

Many more than two goods but let's say you can choose between two goods and – lets just take goods that we we've been doing using in recent videos the two goods that you buy are either chocolate or fruit you can buy chocolate by the

Bar or fruit by the pound so what are going to be your expenditures assuming you spend 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 spend on fruit will be the price of fruit per pound x times the quantity of fruit and so for example so for example if y is equal to $20 a month

20 dollars per month per month and the price and this is we actually will plot this in a second the price of chocolate is equal to $1 per bar one dollar per bar and the price of fruit 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 you all of a sudden you would know what this is you would know what this is and this is you know what the peas are the why and then you can actually graph

One of these quantities relative to the other and what we can do is and let's do that we can graph we can graph the quantity of one 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 and first and if we wanted to graph it I like to put it since I 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 straightforward to graph so let's try that out so first I'm just going to rewrite this without expenditures in between so we have our income our income y is equal to the

Price of chocolate times the quantity of chocolate plus 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 is this one right over here

So if I want to solve for that the best way I could isolate it on one side of this equation so let me get rid of these this yellow part right over here and the best way to do that is to subtract it from both sides so let's subtract the

Price of fruit times the quantity fruit and I can 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 so you see you don't have to just use with these

Numbers you can just see kind of the general result here so 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 minus the price of fruit times the quantity of fruit and 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 by the price of chocolate and then you get and I'll flip the sides you get the quantity of chocolate is going to be

Equal to your income your income divided by the price of chocolate divided by the price of chocolate minus the price of chopped the price of fruit time's the quantity of fruit all of that over the price of chocolate all of that

Over the price of chocolate and we can actually substitute these numbers in here and then we can actually plot what this what this essentially this budget line will look like so in our situation 20 y is equal to 20 the price of

Chocolate is equal to one price of chocolate is equal to one and so this term right over here 20 dollars per month divided by one dollar per bar which would actually give you 20 bars per

Month if you work out the units that so this term right over here just simplifies to 20 and this is actually an interesting term your income your income in dollars divided by the price of an actual good or service you can view this

Term right over here is your real income real real income and the reason why it's called real incumbent's it's actually pegging what your earnings to what you can buy it's pegging it to a certain real goods it's not it's not tied to

Some abstract quantity like money which always has a changing buying power what you could buy for $20 and 2010 is very different than what you could buy for $20 in 1940 but here it's putting your when you divide your income by divided

By a price of some good it's really telling you your income in terms of that good so 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 is 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

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 ten pounds of fruit a month it's tying your income to

Real things not the abstract quantity like money but anyway so this is going to be equal to so let me write it over here my quantity of chocolate is going to be equal to this term right over here is 20 20 and if you wanted to do the

Unit's 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 unit's just like numbers and see how they cancel out 20 bars per month and then

– the price of fruit / the price of chocolate $2 per pound of fruit so it's going to be so the price of fruit is going to be $2 and I actually want to look look at the unit's because that's interesting so it's going to be to let

Me right here so the price of fruit 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 and then we're dividing that by the price of chocolate dividing it by the

Price of chocolate which is equal to $1 $1 per bar bar of chocolate now obviously the math is fairly straightforward we just get 2 but the units are a little bit interesting then you have a dollar in the numerator of

The numerator and a dollar in 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 so this is going to be this is the same thing as

The numerator times the inverse times the inverse or the reciprocal of the denominator right over here so you could say 2 dollars per pound x times the reciprocal of 1 is just 1 times 1 bar per dollar and then the dollars cancel

Out and you are left with – two bars two bars per pound of fruit pound of fruit so we've actually done over here this term right over here it gives us bars bars of chocolate per pound of fruit and it simplifies to two 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 two bars of chocolate because the price you could get two bars of chocolate for

Every pound of fruit so you can view this as the relative price so this right over here is the relative price this is the relative price relative price of fruit in this example let's tell you the opportunity cost to tell you how much

Fruit costs in terms of chocolate bars but regardless that number is straightforward it was just a – so – 2 times the quantity of fruit times the quantity of fruit and this is fairly straightforward to plot so if the

Quantity of fruit is 0 our quantity of chocolate is 20 so this is going to be 20 over here so this is 20 let's just this is going to be 10 this is 15 this is 5 so this is a point on our budget line right over there

And there's multiple ways that you could think about this one way you could say is if you buy no if you buy no chocolate if the quantity of chocolate is 0 what is going to be the quantity of fruit and then you could you could solve this or

You could just say look if I have 20 dollars a month and I'm going to spend it all on fruit I can buy 10 pounds of fruit so let's say that this right over here is 10 so 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 a kind of your typical

Algebra one course you could say look my-my-my in this case if you view this as the y-axis you say your y-intercept or you say my chocolate quantity intercept is 20 and then my slope is negative 2 my slope is negative 2 for

Every extra pound of fruit I buy I have to give up I have to give up 2 pounds of chocolate if you could also view this as the opportunity cost of fruit so you see the slope as we go forward if we buy one more quantity of fruit we're buying one

We're giving up 2 bars of chocolate now one statement I did just make I said every every point of on this line is a possibility and I can only say that if we assume 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 a tenth of a bar of chocolate or on on average especially or we could buy you know a hundredth of a pound of fruit if they weren't divisible if they're indivisible

Then only the whole the whole quantities would be the possibility points but we'll just assume they're divisible especially even if the store only sells indivisible bars of chocolate if you buy 1 bar of

Chocolate every four months on average you're buying two point 025 bars of chocolate per month so even in that way on average almost anything almost anything here is divisible and so this line right over here shows all of the

Combinations we can buy all of the combinations of the divisible goods we can buy if we spend all of our money so that right over there is our budget line that is our budget budget budget line that is our budget line and any

Combination out here is unaffordable on affordable we can't we don't have enough money for that and any combination down here is affordable and actually we would end up with extra money for below the budget line and this isn't all that

Different than what we saw with the production possibilities frontier remember we had a curve that really showed all the we were producing two goods what combinations of goods we could produce anything on that curve for

The production's possibility frontier was efficient anything outside of it was unattainable and anything inside was attainable but inefficient

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