Video transcript
What I want to do in this video is introduce you to theidea of a budget line. Actually, probably isn't a new idea. It's a derivative idea of what you've seen and often in anintroductory algebra course where A, you've gotten acertain amount of money and you can spend it on acertain 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 doit with actual numbers. Lets say your income, yourincome in a month is Y and lets say that youspend all of your money. Your income is equal to your expenditures. Assuming in our little model here that you're not goingto be saving any money. To show how overly simplifiedwe 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 ona two dimensional surface like the screen over here. Obviously, most people buy many more or they at least arechoosing between many, many more than two goods. But let's say you canchoose between 2 goods and let's just take goodsthat we've been doing using in recent videos. That 2 goods that you buy areeither 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 allon 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, actuallywe'll plot this in a second, the price of chocolateis equal to $1 per bar and the price of fruitis equal to $2 per pound. I think these were the prices I used in a per pound of fruit. Then all of a sudden, youwould 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 couldactually graph one of these quantities relative to the other. What we can do is, and let's do that, we can graph the quantityof 1 relative to the other. Why don't we put the quantity of chocolate on this axis over here andlet's put the quantity of fruit on this axis over here. First, if we wanted tograph it I like to put it, since I've put quantity of chocolate on the vertical axis here, I'd like to solve this equationfor quantity of chocolate as a function of quantity fruit and it should make it prettystraight 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 fruittimes the quantity of fruit. Now, I want to solve forthe quantity of chocolate. Let me make that orange so we know that this isthis one right over here. If I want to solve for that, the best way I could isolateit one side of this equation. Let me get rid of this thisyellow 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 substitutethe numbers in first and that might actuallymake it a little bit easier to understand but I like to keep it general first. You see, you don't have tojust use with these numbers you could just see thegeneral result here. I'm going to subtract itfrom 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 incomeminus the price of fruit times the quantity of fruit. This is going to be equalto your right hand side which is just the price of chocolate times the quantity of chocolate. Now if we want to solve forthe quantity of chocolate we just divide both sidesby 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 bythe price of chocolate minus the price of fruittimes the quantity of fruit all of that over the price of chocolate. All over that over the price of chocolate. We can actually substitutethese numbers in here and then we can actuallyplot 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 giveyou 20 bars per month if you work out the units. This term right over herejust simplifies to 20. This is actually an interesting term, your income, your income indollars 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 peggingwhat your earnings to what you can buy. It's pegging it to a certain real goods, it's not tied to someabstract quantity like money which always has a changing buying power. What you could buy for $20in 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 yourincome in terms of that good. You could view yourincome as $20 per month or you could view your income if you wanted yourincome in chocolate bars. You could say my income is, I could buy 20 chocolate bars each month. So I could say, my income20 chocolate bars per month. They would be equivalent to you assuming that you couldsell the chocolate bars for the same price you could buy it and that's somewhat of an assumption. But you could say I havethe equivalent income of 20 bars a month. You could have also done it in fruit. I have the equivalentincome 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 thisterm right over here as 20. If you wanted to do the units, it would be 20 bars per month and you could do a littlebit of dimensional analysis to come up with that. You could treat theunits 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 isequal 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 thatby the price of chocolate. Dividing it by the price of chocolate which is equal to $1 per bar of chocolate. Now, obviously the math isfairly straight forward. We just get 2, but the unitsare a little bit interesting. You have a dollar and thenumerator of the numerator and a dollar, the numeratorof 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 thenumerator times the inverse times the reciprocal of thedenominator 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 2bars per pound of fruit. What we've actually done over here, this term right over here, it gives us bars ofchocolate per pound of fruit. It simplifies to 2 bars ofchocolate per pound of fruit. It's actually givingyou the opportunity cost of a pound of fruit. It's saying hey, youcould buy a pound of fruit but you'd be giving up2 bars of chocolate. Because the price, you couldget 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 isfairly 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 ourbudget line right over there. There is multiple ways thatyou could think about this. One way you could say isif you buy no chocolate, if the quantity of chocolate is 0, what is going to be the quantity of fruit? Then you could solve thisor 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 overhere 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 isgoing to be on our budget line. Another way you could have done this and this comes straightout 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 chocolatequantity 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 asthe 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 onthis line is a possibility and I can only say that if we assume that both of these goodsare divisible goods which means we can buyarbitrarily small amounts of it, that we could buy 10thof 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 ofchocolate every 4 months, on average you're buying .25bars 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 outhere 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 thatdifferent than what we saw with the productionpossibilities frontier. Remember, we had a curvethat really showed all of the if we were producing 2 goods, what combinations ofgoods we could produce. Anything on that curve for the productions possibilityfrontier was efficient. Anything outside of it was unattainable and anything inside wasattainable but inefficient.
