Hi guys, welcome back to In Case of Econ Struggles. Welcome to another micro struggle. Today we're talking about the budget set and the budget constraint.
So what I want to talk about today just very quickly is intuitively the budget set budget constraint. We're going to derive both the budget constraint and the set graphically, and then we're going to talk about how the budget constraint and the budget set change when prices and income change. So again, timestamps as always are below if you would like to jump around, but let's get right into the intuition. And as per usual, when we're talking about intuition. We're going to be talking about Bill.
So here is Bill's latest struggle. So Bill's got 20 bucks. He's going to walk into, I don't know, maybe a bakery. And so in that bakery, he can buy cookies for $2 or coffee mugs for $4.
And so Bill's trying to figure out what bundles of cookies and coffee mugs he can afford. And so I've just written down three baskets. And so basket one, we have four cookies and four coffee mugs.
Basket two, we have three cookies and three coffee mugs. And basket three, we have two cookies and four coffee mugs. So now I'm not concerned with what Bill prefers. I'm just thinking about what Bill can afford right now.
So if we think about basket one, we just think about how much does basket one cost? Well, if he's going to buy four cookies at $2 each, he's spending $8 on cookies. If he's buying four coffee mugs at $4 each, he's paying $16 in terms of coffee mugs, which is going to be $24 total, which is going to be greater than $20. So alas, Bill can't afford the bundle for four, so I'll leave it at red.
Now basket two, on the other hand, with three and three, for buying three cookies, that's six bucks in cookies. Three mugs is 12 bucks in mugs, for a grand total of 18 bucks, which is less than 20. And so we can see that this bundle is definitely affordable for Bill. But on the other hand, Bill does not spend all his money.
He could actually get another cookie, and he has the money to buy another cookie. So because this bundle is affordable but doesn't spend all his money, we're going to say that this bundle is in Bill's budget set. And just for completeness, basket one is just going to be not affordable. And so now after I finish writing this, we'll talk about basket three. So maybe you're already there.
Maybe you're already thinking about whether or not basket three is affordable or not, and whether or not Bill uses all his money to buy basket three. And so let's just go for it. Basket three, two cookies at two dollars a cookie.
Well, that's four dollars. And 4 times 4 for the mugs is 16. That's exactly equal to 20. And so we're going to say that that is on Bill's constraint because it is affordable to Bill, but it's just barely affordable. So that's the difference between the budget set and budget constraint.
So the budget constraint is just barely affordable. And the budget set, you don't have to spend all your money to afford that bundle. So for example, the.00, but it definitely does not use all of his money. So now that we know intuitively kind of what's happening, let's move to a graph. So on this graph if we use those same bundles the first bundle was four and four so right about there.
Second bundle was three and three and we said that didn't use all of Bill's money so that's right here. The bundle that did use all of Bill's money was going to be two cookies and four coffee mugs so that is going to be right here two cookies and four coffee mugs is right here. And so you're like all right what other bundles are either in the set or just barely affordable to Bill?
And so what we're going to say is, well Bill's got 20 bucks, coffee mug costs four dollars, how many coffee mugs could I buy if I spent all my money on coffee mugs? And the answer is going to be five, so that's going to be this point right here. Maybe I'll do this in, maybe I'll do this just in black just to make it sort of easy. Now on the other hand, cookies cost two dollar a cookie, so we can have 10 cookies if we were to spend all our money on cookies.
So if I just go out then it's going to be right here. And so the budget constraint line is going to look something like that, where this green point is on the line. And so we call this our budget constraint. And so you can see, just like we said in the intuitive example, this orange point here is under the budget constraint. It's in the budget set because we can afford it, but we don't have to spend all our money.
Our green dot is in the budget constraint and the red dot is not affordable. So more broadly. Anything on this black line or the budget constraint line is just barely affordable to bill.
Anything outside the budget, so all of these points out here, these are the points that are not affordable at all. So I'm just going to say not affordable. And the points inside the triangle are going to be the budget set. So this is going to be the budget set because this is affordable, but we don't have to spend all of our money.
Now one thing that's useful to know about the budget constraint is that the slope actually has a meeting. So at first, remember that the price of coffee, mugs, or maybe PM is going to be equal to 4, and the price of cookies was equal to 2. So notice, we can think about the trade-off that Bill faces between mugs and cookies because for every mug that Bill gives up he gets to get two cookies. And so the slope of this line, which maybe I'll put in this blue color, so the slope is going to be negative 4 over 2, which is basically just minus 2 over 1 because it's representing the idea that for every coffee mug that Bill gives up he gets to get two more cookies. And if Bill wants to get another coffee mug, he is going to have to give up two cookies in order to get that coffee mug.
So it represents the trade-off that Bill faces in terms of buying cookies versus mugs given that he's spending all of his money. And so that is what we call the slip of the budget constraint. It's just going to be the ratio of the prices.
And if we wanted to write out this equation for the budget constraint, all we would have to say is that P M times M plus P C times C has to be less than or equal to the amount of money Bill has which was 20. Where again this less than or equal to sign we can sort of split out. Where we have less than and equal. So less than is going to be the budget set because those are the bundles that we can afford without spending all of our money. And where it's equal to that is going to be our budget constraint because that is where we spend exactly all of our money.
Now there is a way you can write this in set notation. If you think you need that for class go ahead and leave a comment below. But if this video is making sense so far, please like this video and please also give a comment below.
So again, just to review the constraint versus the set, the constraint are bundles that are just barely affordable, whereas the set are all affordable bundles. And this is going to be more important in a couple of videos when we get to warp and sarp. Now let's get into the fun stuff, which is shifting the budget constraint.
And so what I want to do first is just to say, well, here are five baskets. We have the three baskets from before. and we have four other baskets and i've chosen them in a way that'll make sense when we get to this graph so we know for instance at our original prices at our original wealth or income basket one was not affordable basket two was affordable but not on the constraint basket three was on the budget constraint if we think about basket four we are not affordable either because we're spending four dollars on cookies but 32 on mugs so that is not affordable so maybe i'll just put a sad face here and two and three I'm spending $4 on cookies and $12 on mugs.
So I'm in the budget set. So maybe I'll just put a weak smile in orange. So now let's think about what happens to all of these baskets when we do some shifting.
So first, just to make it easy, let's say that we increase our wealth to $24. Well, we saw before that the cost of basket one was exactly $24 because you're spending $8 and $16. So this is now right on your budget constraint. So I'll write BC.
Now basket B. two was affordable before and it's affordable now. So this is now going to still be in the budget set. Basket three was on our budget constraint but now we're only spending twenty dollars even though we have twenty four dollars. So this is also going to be in our budget set.
Now for basket four we're spending four dollars on cookies but thirty two dollars on mugs. So this is still not affordable. So I'm just gonna write not afford just to make it fit in this little square right here.
And two and three, this was in our budget set before. All we did was get more money, so it's still going to be in our budget set. Again, if any of this is unclear, feel free to put a comment below.
Now let's think about what happens if we halve the price of coffee mugs to two. Well, if we halve the price of coffee mugs to two, then for basket one, we're now going to be essentially buying eight cookies, because both mugs and cookies are worth $2, so we're only spending $16. So this is going to be now.
in our budget set because we can afford it without spending all of our money. Again, just to make it clear, this is going to be where our wealth is going to be back to 20. And same thing here. We're only going to change one thing at a time. So basket two was in our budget set before. And now that the price of coffee mugs are even lower, we're still in our budget set.
Same thing for basket three because we're only spending $12. Now for basket four, we're spending $4 on cookies, but now we're only spending $16 on coffee mugs. So this is now on our budget constraint because it's just barely affordable to us.
And for basket five, we're now going to still be in our budget set because we're spending even less than we did before. Now if on the other hand, we crank up the price of cookies to four, if we only have $20, then what's going to happen is basket one is going to be right back to being unaffordable. So I'll write not afford because we're now going to be spending $16 on cookies and $16 a mug, which is $32.
For basket two, we are going to be spending $12 on cookies, and we're going to be spending $12 on mugs. So this is also no longer affordable. So I'm going to write not afford.
For basket three, again, we're going to be spending $8 and $16 on cookies and mugs respectively. So this is no longer affordable to us either. For basket four, we are going to be spending even more money. So this is not affordable to us at all.
And for basket five, what's going to happen? We are now spending $8 on cookies and $12 on mugs. So this is now on our budget constraint. So again, if any of that was unclear, feel free to put a comment below. But now all we're going to do is translate this onto a graph.
And that's all I've done here. So we'll start with our initial budget constraint. And so our initial budget constraint will go something like this. Okay, now what we need to do is for our white blue guy, what we're going to have is we're going to have that first budget constraint that I showed. which is going to be where we have $24.
So if I have $24, then I can buy one extra mug, and I can buy two extra cookies. And so it's going to look something like this, where the lines should be parallel to each other. And so this red line is actually right here. So this is where we have W equals 24. This makes a lot of sense that we're shifting it out like this. If we haven't changed the trade-out between the two goods, the slope of the line shouldn't change.
Now for number three, where we change the price of coffee to $2, what that's gonna do is that's gonna mean that we can get twice the amount of coffee mugs as we could before, and we can get the same amount of cookies. And so what that's gonna happen is you're gonna have this line that sort of goes like this. It's gonna go way off above my axis like that, but that's roughly what it looks like.
And so what we can see is even though we're back to our initial wealth, because we reduced the price of coffee mugs, We have all these bundles in here that are now affordable to us because of that price change. So anything between this red line, this black line, are bundles that used to not be affordable to us under a wealth of 20 and now are affordable because of the price change. On the other hand, if we crank the price of cookies up to four, then what's going to happen is we can only afford half as many cookies as we could before. So right here, and we can afford the same amount of coffee.
So it's going to look something like this. And what we know from our math is that basket five or the basket in blue should be on the budget constraint. And so I'm just going to do a little bit of sort of pointing and making this dot big so that it actually fits. And so what you can see is everything that we've talked about on the math makes sense on the graph. So now when we make the price of cookies more expensive, all of these bundles in here that I'm highlighting in blue, All of these bundles are basically bundles that we used to be able to afford, but now we can't because the price of cookies went up, and so there are a bunch of bundles that are no longer affordable to us.
And note again, each time the price changed, the trade-off between coffee mugs and cookies changed. That means that the slope of the budget constraint also changed. So again, if there are any confusions or anything doesn't make sense, please leave a comment below.
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