[User] Please transcribe this text

[Assistant] Uh, they're, they're, they're gonna both stop there. So, but they're each gonna stop at the same point where the last ticket is worth the same to each of them, and all the previous football tickets each bought is worth more to each of them.


[User] Please transcribe this text

[Assistant] So, there's no way of rearranging the tickets, because if you did, you'd have to take a football ticket that was worth more than 23 from one of them, and give that ticket to the other one, who by then would think it was worth less than 23. So that's the argument.


[User] Please transcribe this text

[Assistant] And having proved this theorem and given a mathematical form to the invisible hand.


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[Assistant] Whenever you take a verbal argument like Adam Smith and turn it into a mathematical argument, you've understood it better.


[User] Please transcribe this text

[Assistant] You understand why it's true and you also start to understand why in some circumstances it might not be true. So, for 50 years, let's say between 30 and 50 years,


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[Assistant] This was the fundamental argument of economics and economic equilibrium, that equilibrium was a good thing, because it maximizes the sum of utilities. But then, little by little, starting with Irving Fisher,


[User] Please transcribe this text

[Assistant] And a bunch of people at the same time and a little bit later, so Hicks and Samuelson


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[Assistant] Wonder, what kind of crazy utility functions are these, where there's some mysterious good that has constant margin utility of 1?


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[Assistant] In modern terms, how could you justify it? You could say, well, maybe there's a machine where we, uh, can calibrate exactly how happy people are, and we can measure it. So, for example, we can measure how much, uh, how many, how many, uh,


[User] Please transcribe this text

[Assistant] Think of the two goods, x and y, as more or less symmetric. I mean, there's no reason why one has to have some special role.


[User] Please transcribe this text

[Assistant] So let's look at this equilibrium, which we got the same way, you did this on a problem set, solving for equilibrium. Does this maximize the sum of utilities? Well, the answer is, if it did maximize the sum of utilities, at the final allocation, we couldn't gain anything by


[User] Please transcribe this text

[Assistant] So margin utility of A x plus margin utility of B x equals 3 fourths times 1 over 9 fifths, which is 5 ninths. Okay, that equals 15 over 36, which is 5 over


[User] Please transcribe this text

[Assistant] 18, if I've done that right. Doesn't sound right.


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[Assistant] 5 over 12, thank you. That sounds better. And now, but let's look at, uh,


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[Assistant] Okay, which is 10 over 18, which is 5 over 9. So these two numbers are different.


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[Assistant] So clearly, the final equilibrium allocation doesn't maximize the sum of utilities. By switching some goods from A to B, you could make both people better off. So that was a shock to economists.


[User] Please transcribe this text

[Assistant] It meant that the argument they'd relied on, the thing which they were using to persuade, you know, policy people that, uh, the free markets were a good thing, was a, was a false argument. It rests on a premise that's indefensible, namely that there's constant marginal utility and everybody can measure it. So they needed some other definition of, some other way of capturing the mathematical idea,


[User] Please transcribe this text

[Assistant] What makes you think they haven't made another mistake? So now let's give the argument that in a weaker sense


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[Assistant] And later in the course we're going to find that that argument is also too narrow. So my point that I'm going to gradually get to is that economists


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[Assistant] And as you enlarge the view of the world you have, not by making things mathematically simpler, actually you have to make them mathematically slightly more complicated, as you enlarge the view of the world you have, you get closer to the truth and you start to find that the free market isn't quite as wonderful as you thought at first.


[User] Please transcribe this text

[Assistant] And therefore there's a, there's room for government regulation. So it's not, as Paul Krugman argued, that economists are transfixed by mathematics and seduced into


[User] Please transcribe this text

[Assistant] They were afraid of too much mathematics, and by looking too narrowly, they didn't realize what they could have just by, by being mathematically more sophisticated. So it's a failure sophistication, not too much sophistication. So, what did, what did the economists argue? So the chief among them, Hicks, Sanderson, Pareto, and Fischer,


[User] Please transcribe this text

[Assistant] All these people basically came to the same conclusion. They said, well, it makes no sense to maximize, to talk about the sum of utilities. Let's talk about Pareto Efficiency.


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[Assistant] So an allocation, so we begin, so what's the general problem? The general problem is you start with an economy.


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[Assistant] But it could be as many as we want, so let's just say w i.


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[Assistant] And now every, so you start from the economy and you go to equilibrium.


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[Assistant] i equals the summation for all the i's of e i x. Okay, so everyone's fine.


[User] Please transcribe this text

[Assistant] i final consumption of x, plus j final consumption of x, plus everybody else's final consumption of x, plus the sum of the xi's, is equal to the sum of everybody's endowment, and similarly for the y's,


[User] Please transcribe this text

[Assistant] Alright, E-I-Y, and such that everybody is...


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[Assistant] That's the definition of equilibrium. Everybody individually optimizes.


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[Assistant] Supply equals demand. That's how you get the final allocation. That's the definition of equilibrium.


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[Assistant] And now we're trying to argue that's a good thing. Okay, and so what's our criterion? Well, what Pareto and Edgeworth and everybody else decided is let's look at the welfare functions, wA and wB.


[User] Please transcribe this text

[Assistant] So, if you take some allocation, like the original allocation, 2-1 and 1-2, you'll have a utility.


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[Assistant] A welfare of each person. 3 4th log 2 plus 1 quarter log 1.


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[Assistant] That's the welfare of A, and the welfare of B will be two-thirds log one.


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[Assistant] Plus one-third log two. So that'll be some point. Here.


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[Assistant] But then if you look at the welfare at the final allocation,


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[Assistant] So three-quarters log nine-fifths plus one-quarter log three-halves, you know it's gonna be something like this. It's gotta be up here. It's gonna be better for both people. How do you know that? Because A had the choice of just buying his initial endowment.


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[Assistant] That's always affordable and he chose not to do it. He chose to do something else and he was better off and the same as me. So clearly the equilibrium allocation is going to be better than the initial endowment.


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[Assistant] And so the Pareto Criterion, Pareto Criterion,


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[Assistant] We started with an allocation, the E allocation, here, and then we moved to competitive equilibrium here, and everybody was better off. So the equilibrium allocation parietal dominates the initial allocation.


[User] Please transcribe this text

[Assistant] But now the question is maybe there's some other allocation besides the competitive equilibrium allocation that dominates, uhm, that dominates the equilibrium allocation. So the theorem, theorem


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[Assistant] Remember we defined the economy E, I better put E here.


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[Assistant] And formation, got a little too tight here, sorry about this.


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[Assistant] This allocation dominates this one, the competitive equilibrium dominated


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[Assistant] The initial endowment. But maybe there's some other way of rearranging the goods that gets even further out here.


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[Assistant] So what I'm, the theorem says that's impossible. So it says if you start with


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[Assistant] A competitive equilibrium with an allocation XIYI to the economy E, the no allocation


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[Assistant] Like that one, no allocation, x i hat, y i hat.


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[Assistant] Can Pareto dominate it? This couldn't happen. You'd have to be down here somewhere. That couldn't happen provided this other allocation used up the resources that were available. So the sum of the x's in this allocation is the sum of the initial endowments


[User] Please transcribe this text

[Assistant] And the sum of the lies in this allocation is the sum of the initial endowments over the people.


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[Assistant] Of course, if you add more goods to the economy, you can make everybody better off. But using the existing goods in the economy, which is the sum of the EIXs and the sum of the EIYs, there's no way to make everyone better off than at the competitive equilibrium.


[User] Please transcribe this text

[Assistant] So it says literally if P XI YI is an equilibrium for the economy E, which is that, which includes the initial endowment, the no allocation XI hat YI hat, Pareto dominates XI YI if the sum over I of X hat I equals the sum over I of E I X and if the sum of the YIs


[User] Please transcribe this text

[Assistant] Over i, y hat i's over i equals the sum of the e i y.


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[Assistant] Okay, and now there's, that's the theorem. So we've totally weakened


[User] Please transcribe this text

[Assistant] So it might be that if you, if you add, if this is the competitive allocation, if you look at this line, this is same sum of utilities.


[User] Please transcribe this text

[Assistant] has the same utility. So that theorem would say


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[Assistant] Anything else that's feasible had to be down here. The sum is less than that.


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[Assistant] But here we're not saying that. We're no longer saying that. Maybe you can increase the sum. From this equilibrium you could increase the sum by taking something away from A and giving it to B. You could increase the sum


[User] Please transcribe this text

[Assistant] Okay, so that would have been here, you increase the sum.


[User] Please transcribe this text

[Assistant] So you no longer have maximized the sum, but the fact is, to make B better off, you had to make A worse off. And that's the new criterion, Pareto Efficiency. Okay, so that's the theorem, and now we're going to prove it, so that you can see the whole logic of


[User] Please transcribe this text

[Assistant] The free market is just amounts to that. So are there any questions about this? You all supposedly saw this before, but uhm,


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[Assistant] That doesn't mean you really did. So does anyone have a question what's going on? It's a very basic, incredibly important idea.


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[Assistant] Yes, why don't I draw a picture, which is what I was going to do anyway. So let's draw a picture.


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[Assistant] And I'll, I'll bring the theorem back after having drawn the picture.


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[Assistant] So he invented the indifference curve and he invented it in order to draw this picture. It was an incredibly clever picture.


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[Assistant] So he says, let's look at this economy, the one on the right that we start with. We put XA here, and XB here.


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[Assistant] And you notice that A begins with


[User] Please transcribe this text

[Assistant] Okay, now we can draw his indifference curves through that. So his indifference curves, he likes x more than he likes y. So his indifference curves are gonna be sort of vertical, right? He doesn't like to give up x very much. So his indifference curves are gonna look sorta like this.


[User] Please transcribe this text

[Assistant] Alright, and he's not going to choose this. It turns out he chooses...


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[Assistant] Something like here, which is, uhm, he chooses a little bit less of x, 9 fifths,


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[Assistant] Okay, and a little bit more of y. So here's xA equals nine-fifths, which is a little less than two, and xB, he chose, was, uh, I can't read it, three-halves, which is more than one.


[User] Please transcribe this text

[Assistant] Okay, are you with me? Are you asking the question? Does this picture make sense? Okay, so he obviously did better than he did before.


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[Assistant] It's really much, you know, we got another guy to worry about. So how can you put the second guy in the same diagram? Well, let's just finish


[User] Please transcribe this text

[Assistant] Let's just add the total endowment. Where's the total endowment? Well, it's three units of each. So here we've got two, so this is one, uhm, two, this must be three. Okay, so total endowment of good


[User] Please transcribe this text

[Assistant] x is 3, 2 plus 1 is 3, and a good y is 3 as well. Here's 1, here's 2, here's 3. Supposed to be a square. So here's, the aggregate endowment is here. This is the total endowment. 3 of, 3 of each. Okay, but now we can do, we can put b, alright, so,


[User] Please transcribe this text

[Assistant] So this is an extremely clever insight of Edgeworth. He said, look, the total endowment is 3 comma 3. If I tell you that A has got 2 comma 1, that's this point, what's left over B must be consuming. So B is consuming the distance from here to here, which is obviously 1 comma 2.


[User] Please transcribe this text

[Assistant] And if A ends up over here at nine-fifths, three-halves, okay, that's from here to here is nine-fifths and three-halves, since the total is three-three, B's obviously doing, consuming what's left over. So he's consuming this different.


[User] Please transcribe this text

[Assistant] Look at the utilities that A and B got out of this allocation. A got


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[Assistant] Is on this indifference curve. You know what he's doing. B, you know, is coming from here. We know how he's doing. So look at the allocation they got.


[User] Please transcribe this text

[Assistant] You can't make both people better off than that by choosing some other point in the square. So let's look at it from B's point of view. From B's point of view,


[User] Please transcribe this text

[Assistant] For a B, so B's, the axes for B looks like that. So he's gonna have some indifference curve that, uh,


[User] Please transcribe this text

[Assistant] These indifference curves are going to look, uhm,


[User] Please transcribe this text

[Assistant] Okay, so the further out you go this way, the better off B is. So as you shrink what A gets, that obviously means B's getting more, and so B's getting happier and happier.


[User] Please transcribe this text

[Assistant] Okay, so what is the definition of cradle efficiency? Whatever point we pick, like this one to begin with, is there a way of making both people better off?


[User] Please transcribe this text

[Assistant] Well the answer is the competitive equilibrium makes both people better off. Why is that? If you can see it from back there.


[User] Please transcribe this text

[Assistant] Uh, the reason is that A went from this white indifference curve to a better indifference curve over here. B, whose budget line is the same budget line, looked at from his point of view, he went from his indifference curve over here to one here.


[User] Please transcribe this text

[Assistant] Okay, so he went from this yellow indifference curve to a better one. So both of them got better off going to this, uh,


[User] Please transcribe this text

[Assistant] Final allocation that they started with. And of course that had to be true because A, could always have chosen to stay where he was at 2-1. He moved to 9-5-3 halves because it was better. And B, could have stayed where she was at 1-2. She chose to go to 5-6-5-3 halves because it made her better off.


[User] Please transcribe this text

[Assistant] So both of them must be better off at the final allocation than they were to begin with.


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[Assistant] Okay, now what does the theorem say? The theorem says, remember in pictures it says, whatever point we pick here, that defines something for A and defines something for B, then we can look at the utility


[User] Please transcribe this text

[Assistant] The theorem says there's no other point in this whole diagram which puts both A and B on a higher indifference curve for each of them than they were at the competitive equilibrium. So if I read the definition again, it says if I'm at


[User] Please transcribe this text

[Assistant] If I start with an economy there, with the endowments and the utility functions, with the endowments and the indifference curves,


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[Assistant] Okay, then no allocation, that means no point in that box, could make everybody better off.


[User] Please transcribe this text

[Assistant] That new point is in the box, so it makes the sum


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[Assistant] Okay, so what's the proof according to Edgeworth? He says it's all a matter of looking at it the right way.


[User] Please transcribe this text

[Assistant] It's easy to look at the picture from A's point of view. You just look at the x-axis and the y-axis. Okay, this is the y-axis.


[User] Please transcribe this text

[Assistant] You look at, so this is the y axis going this way, this is the y axis, and this is the x axis going this way.


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[Assistant] For A, it's very simple to look at the y-axis and the x-axis for A. A's got his indifference curve getting better and better. Okay, and uhm, B


[User] Please transcribe this text

[Assistant] Okay, so from A's point of view, it's very simple. You look at the endowment, you draw the budget line through it, which is some linear line, and then A picks the best point in that budget set, drawing the tangent, which is the white indifference curve to this point, right here. That's the best he can do.


[User] Please transcribe this text

[Assistant] Now the trick is to see that from B's point of view up here,


[User] Please transcribe this text

[Assistant] From B's point of view of here, things look very similar. Because B's endowment is also at this point. He's got one unit up here of X and two units of Y.


[User] Please transcribe this text

[Assistant] Okay, why does he have that? Because the two together added up to three. So if this is what A's endowment was, then of course what's left over is B's endowment.


[User] Please transcribe this text

[Assistant] So B has to go here. Now what does the budget set look like for B? Well it's also a linear thing with the same slope and it goes through this point. So from his point of view this pink line is also his budget set. It goes from here all the way down to where it hits his


[User] Please transcribe this text

[Assistant] Y axis down here. They've got the same budget set just looked at from opposite sides.


[User] Please transcribe this text

[Assistant] It means that when A chooses this point as the best point, okay, which happened to have been 9 fifths, 3 halves,


[User] Please transcribe this text

[Assistant] Then B is happy to choose what's left over as his best point. So his optimal point, his indifference curve, that's tangent to his budget set, has to be the same point looked at from his origin instead of A's origin. That's the trick.


[User] Please transcribe this text

[Assistant] And so now you realize, okay, so that's the key insight that Edgeworth had, his beautiful picture, and now the point is that if the


[User] Please transcribe this text

[Assistant] The white indifference curve looks like that. Anything that makes A better off has to be on this side of the white indifference curve. And the yellow indifference curve, which is B's indifference curve, anything that makes her better off has to be on this side of the yellow curve.


[User] Please transcribe this text

[Assistant] But the white curve goes that way and the yellow curve goes that way. So there's no point that's simultaneously above A's indifference curve and also above


[User] Please transcribe this text

[Assistant] The pink budget line describes what you can afford to buy. Describes the budget equation, this equation.


[User] Please transcribe this text

[Assistant] So, everyone's gonna choose on the pink line.


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[Assistant] Okay, now what's feasible is they could choose any point that's feasible. This point is feasible. B gets almost nothing and A gets almost everything.


[User] Please transcribe this text

[Assistant] B's never gonna let himself be forced down there because B can choose something on the pink line, so he's always gonna choose something better than that.


[User] Please transcribe this text

[Assistant] Okay, so that's proof number one. I'm just going to give proof number two. So okay, most of you have seen this before, but I'm going to give another proof and then I'm going to ask you, and then I'm going to see whether you really understand what all this means. Okay, so here's a second proof.


[User] Please transcribe this text

[Assistant] Okay, and this is a much better proof. So now, a proof, so I'm going to write this algebraically, the proof of Arrow 1951 and Debreu 1951 separately. Okay, so this is called the fundamental theorem of economics, first welfare theorem of economics. Okay, so all this


[User] Please transcribe this text

[Assistant] Pictorial stuff makes so many assumptions. There are only two kinds of traders, there are only two goods.


[User] Please transcribe this text

[Assistant] I mean, they're never consuming on the boundaries, it's always in this, you know, it's just so many special assumptions and everything's two by two, it's just special case. So let's give a much more general proof of this theorem. So here we've got


[User] Please transcribe this text

[Assistant] Two goods won't play any role, there could be any number of goods, there are two goods only here, but there are many traders, lots and lots of traders. So what's the proof that x hat, y hat couldn't Pareto dominate it? So suppose that we've got, this is equal to summation x hat i, and this is equal to summation


[User] Please transcribe this text

[Assistant] Y hat I. Okay, so we've got another allocation which is feasible.


[User] Please transcribe this text

[Assistant] So could it be that this other feasible allocations of x-hats and y-hats, could that make everybody better off, could that make each i better off than x-i y-i? And this proof is so short and so beautiful, so elegant.


[User] Please transcribe this text

[Assistant] And so convincing that it's mesmerized people now for over 50 years and it prevents them from seeing


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[Assistant] Okay, sorry, so this is line zero. This is just repeating what's the case. W i of x hat high y hat i greater than w x i y i.


[User] Please transcribe this text

[Assistant] Okay, that's, for all I, that's the claim. Could this happen?


[User] Please transcribe this text

[Assistant] I, y, hat, I, and it makes everybody better off than x, i, y, i.


[User] Please transcribe this text

[Assistant] Okay, could this happen? The answer is no. Line one is, if that were true, then it would have to be the case that px times xi hat plus py times y hat i had to be greater than px times eix plus py times eiy. Why is that? Because the budget set of Mr. I says


[User] Please transcribe this text

[Assistant] He spends all his money on x and y in his budget set and does the best he can.


[User] Please transcribe this text

[Assistant] If this thing really makes him better off than what he chose, it can't be that it was affordable, otherwise he would have bought this instead. I mean, bought this instead. So it has to be that for I, this bundle costs more than he can afford. That's why he didn't choose it. So this relies


[User] Please transcribe this text

[Assistant] On agent rationality. Everybody, given his choices.


[User] Please transcribe this text

[Assistant] He doesn't care about the rest of the world, but given his choices, everybody's doing the best thing he can. Okay, now.


[User] Please transcribe this text

[Assistant] Schiller would say people make mistakes, they're crazy, they have no idea what they're doing, they're stupid, there's some guy tells them a story in the bar one night and they totally change their life around, they switch, you know, so he'd say all this isn't true. Alright, but anyway, I, I actually believe that people


[User] Please transcribe this text

[Assistant] So if you don't challenge this, you conclude that if something makes everybody better off, each person must have found this allocation unaffordable. Otherwise, you would have chosen it.


[User] Please transcribe this text

[Assistant] So all I did was that's true for every single person i so I could add this inequality for i plus this inequality for j plus this inequality for all the people and then I use the distributive law to take the sum inside and I have this. Okay, but that is impossible and that's the end of the proof because the sum of this


[User] Please transcribe this text

[Assistant] Xi, x hat i is equal to this sum. That was what we had supposed.


[User] Please transcribe this text

[Assistant] The new allocation just rearranges the endowment differently from the equilibrium and rearranges the x's maybe differently from the equilibrium. It has to be that actually this thing on the left is equal to this thing on the right. It's identical to it. So it can't be greater. And that's the contradiction. That's the end of the proof. So it's two lines. Two lines to prove what, you know, Adam Smith spent 400 pages arguing that the free market is a good thing.


[User] Please transcribe this text

[Assistant] Okay, so this is the basis for the idea that we shouldn't regulate, we shouldn't regulate, we shouldn't regulate. Sounds pretty convincing. So any questions about this? Anything you'd like to say? Are you convinced by this?


[User] Please transcribe this text

[Assistant] Okay, so what are some reasons you don't believe this? And some obvious reasons.


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[Assistant] So that, I'm glad you asked that question, because it, no, no, it betrays a misunderstanding, so I'm glad you asked that question, so.


[User] Please transcribe this text

[Assistant] Okay, so this theorem is correct, given the assumptions. Now, he made a standard, uh, okay, so I, there's a little bit of a mistake in the reasoning he made. I'm going to repeat his question, change it a little bit, so maybe he'll deny this is what he was saying, but I believe what he just said is,


[User] Please transcribe this text

[Assistant] What really happens in a lot of regulations is they come in and they change the prices. They tax the good, they prevent people from trading so much that the price changes, they subsidize something. Something happens so that you get different prices, and of course at those different prices you get a different allocation.


[User] Please transcribe this text

[Assistant] And then he said, that's all true so far, and then he said, because they're at different prices,


[User] Please transcribe this text

[Assistant] Uhm, you know, you can't compare the original equilibrium allocation to the new equilibrium allocation. Well, that's where he shortchanged this proof.


[User] Please transcribe this text

[Assistant] This proof says, however you get to a new allocation, x hat i, y hat i, maybe it's because the government has intervened and changed all the prices and done a bunch of other stuff, but in the end, after all that intervention, the government is gonna get you to a new allocation, x hat i, y hat i, and that new allocation can't be better than the original one. Okay, so.


[User] Please transcribe this text

[Assistant] So that's the force of the argument. No matter what the government does, it's gonna, in the end, the upshot is a new allocation, x hat i, y hat i. We don't have to think about how it got there, that's where it got, and according to this argument, it can't be better. So the argument is correct.


[User] Please transcribe this text

[Assistant] Okay, so one argument you could say is that, alright, it may be in the equilibrium that A, okay, so who's better off here? You notice that this equilibrium, A ends up with more of everything than B does.


[User] Please transcribe this text

[Assistant] Okay, so A, so you could say this equilibrium is not very socially desirable, because A ends up with more of everything than B does, or at least as much of everything as B does, and that seems unjust and unfair, and so we don't like the allocation, and so maybe we should move something from A to B. So that's an argument on the basis of justice. It says we can find juster allocations,


[User] Please transcribe this text

[Assistant] Because that, you know, serves your, your desire for fairness. But that doesn't say anything about how efficient the allocation is. It's still true that the original allocation was efficient in the sense you couldn't make anyone better off.


[User] Please transcribe this text

[Assistant] So the economist would say, in order to hurt A and help B, what you should do is take some of A's endowment away from him and give it to her and then let them trade to a new competitive equilibrium where she began with more than him. So the just thing to do is to rearrange


[User] Please transcribe this text

[Assistant] An argument from fairness, but it doesn't, it doesn't interfere, it doesn't contradict the economic argument that fair or unfair, it's still efficient. We've done as well as we could making everyone as happy as we can. Maybe we can hurt someone and help someone else, but we can't help everyone at the same time.


[User] Please transcribe this text

[Assistant] No, it's got all those assumptions. So there's something about the model that's too narrow, and you should, and this should be easy, and you're gonna tell me obvious things where it's too narrow, and then I, and then, so I'm gonna just say where we're going. You're supposed to be telling me now obvious ways this is too narrow. I'm gonna say you're right, you're right, you're right, but finance is different. We don't have to worry about those problems, supposedly. So, go ahead, back there, you. Okay, so one critical


[User] Please transcribe this text

[Assistant] Shortcoming of this argument is that they're externalities. Suppose that uhm, x is cigarettes.


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[Assistant] And A, smoking, buying and smoking more cigarettes makes B sick.


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[Assistant] Okay, so that's not part of this. Why is that not part of this? Because B's utility depends only on what B's consuming, not what A's doing. Okay, so there's no place in this model. It's too narrow to include externalities.


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[Assistant] So it doesn't capture the fact that A might be making, he by smoking might make her sick by smoking so many cigarettes and that's not in here. And if you put the idea that utility, B's utility, her utility might depend on what he's doing and not what just, not what she's doing.


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[Assistant] The theorem won't be true. So there's a reason to tax pollution and to tax cigarettes and all those things because of externalities. Okay, so that's a fundamental problem.


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[Assistant] Well, we're assuming diminishing marginal utility, but they're different. Each person has a different indifference curve. So yes, we assume that they're all


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[Assistant] Curved away from the origin, so we assume diminishing margin of utility.


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[Assistant] But I think, you know, that, that I'm almost prepared to believe diminishing marginal utility. You know, it's not literally true, you know, like two tickets are better than, you know, uh, going from


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[Assistant] Moving from one ticket to two tickets may be a huge gain in utility because now you can bring your friend along and going to third, a third ticket, uh, you know, going from, getting one ticket might not be worth much because you don't want to go alone. Getting the second one may add a huge amount of utility. That's an increasing margin of utility.


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[Assistant] But eventually, after you've got enough tickets, it's going to be diminishing marginal utility. So I believe on the whole that diminishing marginal utility is not such a bad assumption, but this clearly relies on that. So you're right, but I don't think that's a critical problem. What else is there? Yes? What?


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[Assistant] Okay, credit work. Now we're starting to talk a little bit about finance. So just hold that thought for one second. Anything else you can think of besides externalities that's terrible for... Okay, so...


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[Assistant] Alright, so let me transition this way. So of course people noticed externalities. Another kind of regulation is


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[Assistant] Perfect competition. You know, we assumed everyone took the prices as given. There were enough, there was no monopolist, you know.


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[Assistant] The competitive equilibrium to allow for perfect competition, so we assume that already, that's one place for regulation, and a second place is because of these externalities.


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[Assistant] So for 100, you know, 50 to 100 years, everybody has accepted those two arguments. Clearly, there's a place for regulating free markets.


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[Assistant] Insuring free markets and stopping trusts and monopolies and for stopping externalities. But then people said


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[Assistant] Finance, okay, is different. As long as, you know, there aren't any externalities. You know, you're just trading stocks and bonds. There's no pollution. Nobody's gonna get sick because someone else has a stock or a bond. Some people might be jealous.


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[Assistant] So we're not really going to take that into account. We don't, we don't want to honor those kinds of feelings of jealousy. So the argument seems to be when you get to finance, you don't have to worry about externalities, and yes, you have to worry about perfect competition, but once you've got perfect competition, that's the end of the story.


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[Assistant] So the rest of the course now is gonna be, if you forget about externalities,


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[Assistant] Okay, because you forget about pollution, because you know, we're trading pieces of paper, they're not polluting anything. Why shouldn't


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[Assistant] The financial markets be efficient. Why doesn't this argument apply to the financial markets? And one of the critical financial markets is the credit market.


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[Assistant] Can you understand the credit market on just these terms and therefore argue that the market's efficient?


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[Assistant] Comment is a good introduction to the next part of the course.


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[Assistant] That seems to directly involve finance and what we've been talking about. So where does the finance come in? So Irving Fisher was the person


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[Assistant] Uh, he, you know, was a superstar Yale undergraduate. He invented, uh, anyway, he had a, he had a, he graduated first in his class from Yale.


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[Assistant] He decided that he was a math major. He decided that he wanted to go


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[Assistant] He wanted to build an economic, a financial equilibrium model


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[Assistant] There were no economists there at his time. We're talking in the late 1800s.


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[Assistant] And so, he went to Gibbs, the famous physicist, maybe, you know, one of America's most famous physicists of the time, and said, can you


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[Assistant] Can you, uh, advise an economic, you know, an economics PhD?


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[Assistant] And Gibb said, well if it's mathematical enough and has a model in something, yes I can do it. And so,


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[Assistant] Fischer's, uh, PhD dissertation was basically reinventing


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[Assistant] This general equilibrium model that I've just described and then making a machine to calculate equilibrium with water and pumps and water seeking its own level and solving for equilibrium. And we had this machine up until the 1970s in the Cole Foundation when it got stolen and, uh, some


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[Assistant] Some engineer named Srinivasan about ten years ago was ready to rebuild the machine because we have the dissertation where it explains very carefully how Fischer built his machine and he was going to rebuild it.


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[Assistant] And then he was offered ElBaradi's job. ElBaradi is the guy, the nuclear inspector who, you know, goes to Iraq and says they're building nuclear stuff and not building anything. So he was offered that job and so he left Yale.


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[Assistant] And then decided not to take the job after all, so he still got El-Baradi in the job. But anyway, he left Yale, and so he never got the machine built. He was a great guy. And, uh, anyway, so Fisher, uh, after, uhm,


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[Assistant] Fischer, after writing his dissertation, stayed at Yale as an assistant professor and got tenure. He became the most famous economist probably in the country for a while. Uhm, he


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[Assistant] You know, in retrospect, clearly the most interesting model of finance of its time.


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[Assistant] Ten million dollars, uh, for his company in the early 1900s.


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[Assistant] And, uh, he was a friend of Roosevelt, and he kept advising Roosevelt, uh, during the Depression.


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[Assistant] But Fisher's ego seemed to be so big that he, he just, I mean, you know, Roosevelt says, I'm very glad that the good professor has made such worthy recommendations and will certainly take them seriously. And then Fisher writes back and says, I'm glad you're going to follow what I suggested. And so, so anyway.


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[Assistant] Fisher convinced that the economy was going to turn around.


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[Assistant] So he had to go to his, uh, he had to go to his wife, who was very wealthy, and borrow her money, and they lost that money. Then he went to her parents and borrowed all their money and lost that money. And finally he'd lost everything of his own money, his wife's money, and his wife's parents' money.


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[Assistant] Set a bad precedent for economists at Yale because whenever any economist at Yale has financial advice to give,


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[Assistant] Someone always quotes Irving Fisher's 1929 line that the economy had reached a permanently high plateau.


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[Assistant] But anyway, despite all that, he still, uh, his theories are well worth, uh, studying.


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[Assistant] He was famous for a few other things. He's a little bit, Schiller is clearly trying to imitate Fisher.


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[Assistant] Fisher wrote a huge number of books. 50 books about everything. So he got tuberculosis.


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[Assistant] And he survived and he wrote a series of books on good living and good health and how to combat tuberculosis. So he said you had to exercise, you had to get fresh air. These were huge best sellers. And uhm, then he said that drinking was a terrible thing.


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[Assistant] So he did an experiment in class where he would have his students do push-ups and he'd give them all a martini. Then he'd count how many push-ups they could do and then he'd give them a second martini and count how many push-ups they could do. And he found that there was a 10% reduction for each martini and so he said that prohibition, since the average business person had two drinks


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[Assistant] At lunch, prohibition would increase output by 20%. And so he's one of the, he was one of the leading, he was one of the leading proponents of prohibition.


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[Assistant] So in that, after the stock market crash, he helped form the Coles Foundation. Uhm, so he, so he and a bunch, so what, so Coles, uh, who was a Yale undergraduate, uh, had run a very famous, uh,


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[Assistant] Uh, macro forecasting company in the 1920s.


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[Assistant] And in 1929, the stock market crashed, and he, there were a whole series of these. They sent newsletters, same kind of stuff they do now. He, unlike all the others,


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[Assistant] After the fact, realized that he hadn't anticipated anything about the crash, much like nobody anticipated it now. And he was so embarrassed, he went and collected not only his old recommendations himself, but all his competitors. And so he published a famous paper in which he argued that economists had no idea what they were talking about, the fact that they were frauds.


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[Assistant] And so he went to his old economist Irving Fisher at Yale and said,


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[Assistant] Look, I just believe that economics without mathematics has no meaning.


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[Assistant] I want to start a mathematical wing of economics, and I've got a lot of money.


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[Assistant] So, with Cole's money, Fisher started the Econometric Society, uh, which is the most famous mathematical society in economics.


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[Assistant] Econometrica, which is the most famous journal in economic


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[Assistant] And the Coles Foundation, which they moved, which they started off in Colorado Springs because Coles, uh, Fisher set off.


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[Assistant] The weather's so good there, everyone would want to go there, and it's good for them. So nobody would go there, so they had to move it to Chicago, where the Tribune was.


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[Assistant] And it was in Chicago from 1930 to 1955, and then in 1955 it moved to Yale.


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[Assistant] And since then, the Coles Foundation has been, uh, at Yale and you'll hear a lot more about it later. But anyway, so Coles, with that background, he set out to figure out how to turn economics, in fact he invented economics, this model of equilibrium, to study finance. But so far we don't have any finance in the model. So how can you put it in? Well, what is missing? What do I need to put in the model in order to have


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[Assistant] Okay, so the first thing, of course you may have read ahead in the notes, but that's a brilliant insight, if it's yours.


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[Assistant] You need time in the model. So far we just had


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[Assistant] You know, apples and oranges or something being traded. So the first thing, as Fisher said, you need to put time in the model. What else is critically missing in this model? One other thing, the major other thing, yeaah. What? Okay, well that's good, there's risk. That's missing entirely.


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[Assistant] So Fisher actually couldn't figure out how to put that in the model, but we're gonna...


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[Assistant] We're gonna get to that. So what else is Fisher? What else? That is very good. What else is missing? Yep.


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[Assistant] People always care more about the present than the future. But what other fundamental thing about, what is finance all about? When you think about finance, what do you think about? Yeaah.


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[Assistant] Okay, see, think about credit, as she said. Credit. Interest rate.


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[Assistant] All this has to go in the model, but what fundamental object, what do you think, the first thing you think about finance, what do you think of? Yeah. Money. Okay, so, that's not what I had in mind, but I'm glad you think about money. That's what most, he's right, that is what most people think about.


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[Assistant] So Fisher did provide his theory of money, and that's what he was talking to Roosevelt about all the time, but we're not gonna...


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[Assistant] Loaning funds, well somebody said credit, that sort of loaning funds and interest rates. Collateral, that's something, I'm going to add that. Thank you. What a brilliant idea.


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[Assistant] What, what, what, uh, what basic thing, come on, yeaah.


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[Assistant] Wealth, okay, that's in the model. Where is wealth? It's already there. So you know, when you take your endowment and you multiply it by the prices in your budget set, that's your wealth. So wealth is already, we've got wealth.


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[Assistant] Capital, okay, so capital and what's the, and what is capital? What do you mean by capital? If it's money, I already told you that's not quite going to be there.


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[Assistant] Contract, okay, so we're getting there. Contract. Assets. Okay, so assets. That's the main thing. So I'm going to write it over here. Assets.


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[Assistant] Okay, after all, the stock market is about, I mean, finance is a lot about the stock market, bond market.


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[Assistant] Mortgage markets, those are all assets that you buy that pay money, pay dividends later. So assets are things that pay dividends at a later time. That's the critical thing that's missing. And when we talk about finance, we want to know, what should the value of the stock market be? How much should a mortgage be worth? And it'll, and, and, uh,


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[Assistant] How horrible that would be, would mean I also wasted three lectures.


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[Assistant] Fisher said, you don't really have to change very much at all.


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[Assistant] So Fisher said, you know, time isn't such a big deal. We've already got two goods. You know, these old, these, these crazy marginalists before me, they had good acts, which was interesting, and this very boring good why, and so, you know, things, it's not so clear how you can have


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[Assistant] Uh, time and everything in the model, but I've already got two interesting goods, x and y. They're both, you know, entering on, on the same, uh, you know, with the same symmetric properties. I mean, no, neither one of them seems more interesting or more important than the other one. So in fact, maybe they're the same good. It's just x and y is the same good. It's one, one comes one period later.


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[Assistant] Now next class we're going to talk about you're not going to value them the same amount. The utility for x and y won't be the same. So here a obviously thinks more of x.


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[Assistant] Then he does a Y. And by the way, B, she also thinks more of X than she does about Y. So why is that? Well if Y is the same as X, but a year later, people are impatient. So the reason why


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[Assistant] A might like X better than Y and B also is not because they're different goods but because they're impatient. So we're going to come to that later. So we can have the idea that there are two different


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[Assistant] How can you put, so I'm gonna come back to that in the next class. What about assets? How can you introduce assets into the economy?


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[Assistant] Well Fisher says, what is the essence of an asset? The essence of an asset is that


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[Assistant] And we're not going to talk about money, so we're imagining these are all real assets. Uhm, so they're going to deliver goods in the future.


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[Assistant] So assets, the definition of an asset is an asset, remember we have to make it mathematical.


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[Assistant] The whole Fisher started the society to make economics mathematical. So an asset is literally something that delivers


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[Assistant] Goods in the future. So an asset is defined by its payoff.


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[Assistant] Alpha, an asset Alpha is defined by its Theon.


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[Assistant] dx and dy. So it's going to pay a certain amount of x and a certain amount of y. That's all there is.


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[Assistant] Now, we're gonna ask, do people actually know what the payoff of Y is gonna be? If Y is in the future, that means it hasn't happened yet. Do we really know what Y is gonna be? And so Fisher, so we have to, and so, you know, maybe we know less about one asset, more about another asset. So maybe one asset's riskier than another asset.


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[Assistant] Someone said risk. Obviously that's going to be a very fundamental question. But before we come to risk,


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[Assistant] So the asset is no different from what its dividend is going to be. Maybe it's paying something today and also it's going to pay something tomorrow. So assets are defined by their dividends. Nothing more than that. So the model now becomes a model where we have the A, W, I and then we have, uh, W, I and then we have


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[Assistant] The goods as before, the endowments of the goods, but we've added a new thing, which is we've added the acids, D, alpha, X, and D, alpha, Y. So the alpha's over all the acids, and we have to add everybody's ownership of the acids to begin with.


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[Assistant] So we have to have theta, I'm gonna call it theta bar alpha. Theta bar I alpha.


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[Assistant] So that's the economy that we're going to start with.


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[Assistant] Okay, so we've made the model much more, we've made the model a bit more complicated. We've started what we, where we did before with, uh,


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[Assistant] Utilities and endowments of goods. Now we're adding assets, which pay off more goods. And of course people begin owning trees and stuff like that. So a tree is an asset. It's going to pay apples later on.


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[Assistant] And people, various different people own various different trees.


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[Assistant] So we've got to the old economy, we add something new. And now Fisher's


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[Assistant] Ingenious insight was that if you think about this, which we will for a little while,


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[Assistant] You'll be able to discover that you can simplify this economy back down to the economy we began with.


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[Assistant] What is px over py? Suppose they're both apples. So x is an apple today, y is an apple next year. What is px over py?


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[Assistant] It says how much an apple is worth today relative to how much an apple is worth tomorrow. So this is going to turn out to be one plus the real interest rate, Fisher would call it.


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[Assistant] It is, uh, you know, 20% more than PY. What it means is by giving up one apple today, you can get 1.2 apples tomorrow. That's just what happens when you put money in the bank. You give up something today, you get exactly the same thing back next period, but more of it.


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[Assistant] So you see just by having time and the prices like we had before, you're gonna see very quickly how interest, assets, everything is gonna start coming into play very quickly.


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[Assistant] I didn't put the problem set on the web yet because I wasn't sure how far we'll get, but I'll put it on in a couple hours and we'll start again on Thursday.


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[Assistant] I'm with, uh, Stephen Curry, and, uh, Stephen, you told me yesterday that you would be happy to play anywhere, but in your heart of hearts, where would you like to get selected tonight?


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[Assistant] We will discover all of the most popular components and concepts in this section. So creating a logical application.


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[Assistant] In this video we are going to take a look at solving a simple Olympiad programming task. So here is the task.


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[Assistant] It turns out that a fraction is said to be regular if its numerator is smaller than the denominator and irreducible. If the numerator and denominator are relatively prime, so a team is very fond of mathematics and therefore at home, it solves many problems.


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[Assistant] So we got here n, and this is the range of the n value here. So output which will continue two numbers, the numerator and denominator of the fraction phones, must be separated by a space.


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[Assistant] Next step is check a and b for even numbers in the while loops. Then we have to check this a plus b to n. Then if it's true, we have to break the looping. So here the a is denominator and b is numerator.


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[Assistant] So here we got simple Delphi console application. Here we can declare the variables. So, i, it's for iteration, a, b, d, numerator and a.


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[Assistant] So, here, and check the a plus b for n. Then we can break all of the looping here.


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[Assistant] So if you want to learn all of the steps, all of the actions, all of the divisions, subtractions here, you can simply toggle the breakpoint to this while loop or another while loop here. Then you can simply run with debugging this application.


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[Assistant] In this video we're going to take a look at solving algorithmic problem with looping. So here is the task.


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[Assistant] The task is about finding the prime numbers until given value n. So what is a prime number? A prime number is a whole number greater than 1.


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[Assistant] So step 2, we have to write nested for loop. And the first looping starts from 2 to n. So why we should start from 2 the looping? Because the first prime number is 2. We cannot start from another value, for instance 0 or 1.


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[Assistant] Then we have to assign true value to prime variable every time in the first loop. Step 3.


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[Assistant] The second for loop starts from 2 to square root of i variable. Then, if imodule's k equals 0, then the prime variable will get false value, and we have to break the looping there.


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[Assistant] So here is the console application. We can start by declaring the variables var n, i, k integers, then the prime boolean


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[Assistant] Then, it will work. So then, we have to check the iModulesK. If iModulesK equals 0, then tag in here, prime gets false value, and we can break the loop in here.


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[Assistant] 100, and here we got all of the prime numbers from 2 until a 100. So that's it.


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[Assistant] Excuse me baboons, could you tell me what your most favorite thing is?


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[Assistant] I think we're pretty high. I really don't look forward to looking down. I'm looking down. I don't think you should because it's really scary.


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[Assistant] Alright, so last time, uh, we began, or maybe two times ago we began a discussion of various


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[Assistant] vocabulary and facts. You have to know about the markets if you want to think about finance.


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[Assistant] Today we're gonna deal mostly with the most important one, the most basic one, the yield curve. And last time we introduced this word yield.


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[Assistant] Now, yield is an extremely common expression in


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[Assistant] Finance and it turns out not to be that, uhm, well defined often or that useful.


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[Assistant] And, but the word is so important is when used so often that it still hangs around even when probably we should use different concepts.


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[Assistant] To look at an investment and without paying any attention to the market or anything outside the investment just looking at the investment itself.


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[Assistant] Try to assess, give a number, quantifying how attractive the assessment was. Attractive the investment was.


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[Assistant] So, we said you could apply that to a bond, it has cash flows, you could apply it to a hedge fund that's taking in money and paying out money. And, uh, the formula we came up with said that if the cash flows are given by C1


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[Assistant] Then, uh, and its price is some minus, uh, you know, some price P0, maybe it's a negative cash flow, so C.


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[Assistant] So some of these cash flows might be negative and some of them might be positive, then we should just look.


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[Assistant] At the number y such that discounting all these things at rate y gives you zero.


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[Assistant] The why that did that was what we defined as the yield of the investment.


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[Assistant] So we saw that that had some advantages. For example, in a hedge fund, if you just look at the rate of return it makes on its money every year, that doesn't take into account that some years it's got a lot more money. So if those were the years that lost money, and the years when it hardly had any money were the years that it made money,


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[Assistant] Uhm, just taking the average, the multiplicative average of all those, geometric average of all those


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[Assistant] Well, the yield also gives a somewhat misleading figure, and I don't want to spend too much time on why it might be misleading.


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[Assistant] Uh, but I'll give you just an example. Suppose that the cash flows happen to be 1, minus 4, and 3. Now what's the yield to maturity? Well, there are two of them. You could have y equals 0, because


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[Assistant] Minus four over one plus zero plus three over one plus zero is just one minus four plus three. That equals zero. So the yield to maturity of zero percent, uh,


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[Assistant] The yield of 0% makes this have present value 0, but also I could try y equals 200%. And then I'd have y


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[Assistant] And I'd have minus one, or plus one, minus four-thirds, plus


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[Assistant] 1 over 3 squared is 1, uh, 3 over 9, so it's plus 1 third, so it'll be 1 minus 4 thirds


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[Assistant] Plus one third, which also equals zero. So is the yield to maturity of the internal rate of return zero percent or two hundred percent?


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[Assistant] It's ambiguous. So yield to maturity can't be the right way of doing things. To go back to the hedge fund example,


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[Assistant] You know, the hedge fund was taking in money, paying out money, taking in more money, paying out money.


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[Assistant] And we calculated the yield to maturity. Well suppose that there was some period, you know, here at which everyone had taken all their money out. So the hedge fund wasn't actually doing anything for a bunch of years, maybe for a long time, and then it started up and, and you know, took money in and paid money out and stuff.


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[Assistant] Well, because the gap in time was very long with nothing happening, if you take a positive Y, the stuff that happens in the second incarnation of the fund is hardly going to be making any difference, because by that time it will all be discounted a lot.


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[Assistant] The yield will depend too sensitively on, uh, stuff early rather than stuff late.


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[Assistant] And so, uh, again you get into trouble yielding just, uh, yield to maturity.


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[Assistant] So that can't be the right thing to do, even though people have done it for years.


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[Assistant] An investment where there's no doubt about what the cash flows are going to be, what would he say you should do to evaluate the attractiveness of it?


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[Assistant] What's our lesson, our mean lesson from Irving Fisher? What would he say? Yeaah. Well, let's say there


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[Assistant] Cash flows, so it's money. Money that you're gonna get coming in and out. Yea, he'd say...


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[Assistant] Actual, you know, potatoes every time, you know, apples each time, you know, deflate by inflation, and then do what? With the numbers.


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[Assistant] Okay, he'd say just look at the present value of all these things. So of course to do that you'd have to know what is the market rate of interest with which to compute the present value. So Fisher would say it's ridiculous


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[Assistant] What the, uh, present value of all the cash flows is. So we're going to now, uhm, do that a bunch of times. Okay, for the rest of the class and see what that means.


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[Assistant] How do you know, so we have to begin, the first, the two thirds of the class is going to be spent on the question, how do you know what the market rates of interest are?


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[Assistant] So how do you know what the market rates of interest are? How could you find out?


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[Assistant] What, I mean, what the market rate of interest is. What would you do to find it out? Yeah.


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[Assistant] And if you looked in the newspaper, say, could you find in the newspaper, what would you find in the newspaper? Yep.


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[Assistant] Okay, and so how, so alright, so yes, you try and look at riskless investments like government bonds where there can't be any default.


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[Assistant] We've got what they always used to say. It can't be any default on an American, uh,


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[Assistant] Promise, America's never broken, government never broke a promise.


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[Assistant] Uh, and they can always print the money, so presumably they don't have to break a promise. So they're just promising money, which they can print. So why should they ever break their promise?


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[Assistant] What would you find if you opened a newspaper? You would find for different maturities,


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[Assistant] Uhm, used to be up to 30 years for different maturities, you would find the


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[Assistant] The yield of various government bonds, of gov bonds. Why do they quote the yield? Well, that's just because, you know,


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[Assistant] 100 years ago people started using the idea of yield and so the vocabulary has been kept even though


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[Assistant] It's not the best, uh, way of describing what's going on.


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[Assistant] For instance, let's just look at some of the yield curves you might have seen over the last ten, or nine years, almost ten years, since December 2000.


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[Assistant] You would see that in December 2000, the yield on the one-year bond, you know, or the short yield,


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[Assistant] This is on a log scale, so this is, you know, 3, 6, 12. Okay, so the shortest bond,


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[Assistant] Uh, usually have lower yields than the highest bonds.


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[Assistant] But sometimes, like in December 2000, the yields are almost all the same. It's called the flat yield curve. Other times, like, uh, now,


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[Assistant] We're in this light blue one here, like now.


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[Assistant] The short bonds have very small yields.


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[Assistant] Okay, so you get the yield on every single bond. Now what do you notice about this picture, by the way? They can be very different at different time periods.


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[Assistant] So, in December 2000, the interest rates were really high. The yields were 6%. I'm talking yields so far. We haven't talked about interest rates. We have to figure out what the interest rates are. But anyway, they're obviously going to be connected. So the yields were very high in December 2000.


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[Assistant] And they got much lower in December 2008 and they stayed very low. So why are they so low now? What got them to be so low now? Yep.


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[Assistant] The Fed flooded the economy with money. It wanted to drive the interest rates down to zero.


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[Assistant] It's, it's changed the yield curve. So you notice that the yield curve now, December 2008, was this blue one, so the Fed in the crisis of 2008, you know, was


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[Assistant] Terrified, and it dropped the interest rate almost to zero, virtually zero, and it's kept it there because from December 2008 till now, we're at the end of, we're almost, we're October 2009, okay, September 30th, 2009, a long time has passed


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[Assistant] From the dark blue to the light blue line, and the, and the rate has been kept fixed there.


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[Assistant] But in the intervening time, the long rates have started to go way up. Now, why might that be the case?


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[Assistant] That could be one reason, and could there be another reason?


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[Assistant] Okay, so those are the two reasons. So they somehow know that, and some of you have no idea how they could possibly be thinking that.


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[Assistant] And so we're, I'm going to explain why, what information is, is there in the different yield curves. Okay, so.


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[Assistant] The point is that every morning, every single financial analyst wakes up


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[Assistant] And sees these yield curves, you know, consults the market and sees where things are trading and can produce a yield curve like that.


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[Assistant] Do, uh, an example here. So I'm going to make up an example which is very easy to compute. So, let's try this one. Okay, so suppose if I'm reading here at the top, let's just say that, uhm,


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[Assistant] You've got a bunch of different bonds, a one year bond, a two year bond, a three year bond, a four year bond, and a five year bond.


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[Assistant] Uh, let's say they were, you know, often they might, so I'm assuming here that they're all issued on the same day. So,


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[Assistant] They're issued with different coupons. Let's say they were all issued today. We'll come back at the end. You know, obviously the Treasury doesn't issue new bonds every single day. So how does this change when you arrive on a day when they haven't issued things? But let's just keep it simple and suppose that today the Treasury's issued


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[Assistant] You know what coupon they're going to set. So they set the coupon. So let's say the coupon they set.


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[Assistant] Was a dollar for the one year bond, two dollars for the two year bond, three dollars for the three year bond, four for the four year bond, five for the five year bond. It's easy to remember. That's why I chose those numbers.


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[Assistant] And the face value, let's say, is always 100. So why did they set those coupons?


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[Assistant] Well, because given how much they want to sell, they're picking the coupon hoping that the price turns out to be close to the face value. So let's say when they actually market these,


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[Assistant] And uh, supply equals demand and equilibrium, the prices turn out to be 100.1.


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[Assistant] So 100.5 is the price the market's paying for the 5 year bond, and if the coupon is 5 and they pay coupons once a year, they actually pay


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[Assistant] They may pay twice a year, but let's say they pay once a year for simplicity. You're going to get 5, 5, 5, 5, and 105 the last year.


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[Assistant] Okay, so those are the bonds. Now the newspaper's not telling you any of that. So you're sort of losing that information.


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[Assistant] Uhm, so you don't actually know that from reading the newspaper. So then, what do you know? Uh, you know that you know the, uh, you know the yields on all these things. Okay, so.


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[Assistant] Okay, if the, knowing the yield, you could figure out what the price of each of the bonds is, or in fact the way that they calculated the yield that the newspapers reported. How did the newspapers get the yield? The newspaper said, well, for the four year bond,


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[Assistant] 4 divided by 1 plus the 4 year yield, the 4 year coupon bond yield plus 4 over 1 plus y of 4 squared.


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[Assistant] In this case, because these numbers are all positive, obviously it's monotonic in Y of 4, so the unique Y of 4, which you can use to solve this equation, price equals the discounted value at that yield.


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[Assistant] Okay, so that's how the newspapers, the reporter, that's how he got all the yields to show you that graph. He looked at the market or called the bank or something like that, had a computer screen, you know, talked to his friends on Wall Street, he knew the price of all the bonds, he knew the coupons of all the bonds, and then he produced the yield for all the bonds.


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[Assistant] So the yield, as I say, it's just, that's the word that everybody uses, but really the information that you want to deal with is the price and what did the coupon actually pay.


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[Assistant] Okay, so that's what you know. Everybody knows this. Every day, the information I've given, every morning, maybe every few hours people will update it. They'll look at what are the coupon bonds paying and what are the prices. The things that's changing, you know, from hour to hour are the prices.


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[Assistant] But we're taking a snapshot at the beginning of the day and looking at the prices. So now, we've got prices of bonds, which I'm going to call capital Pis, Pi 1, Pi 2, Pi 3, Pi 4, and Pi 5.


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[Assistant] The most important thing to do is find the interest rates. So what has this got to do with interest rates? Well, if you modernize Fisher a little bit,


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[Assistant] The most important thing, he didn't put it this way, but this is really what he must have meant. The most important thing to do is find the prices of the zeros.


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[Assistant] Today's money price. Today's money price.


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[Assistant] Today's money price, one dollar at time two, pi three,


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[Assistant] Today's money priced at time 3 and pi 5. Today's money priced at time 5.


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[Assistant] Okay, now why do you want to find these things? Because once you know these things, you'd be able to value any investment. That original investment that we talked about, which maybe disappeared, this one, uhm, haha, it disappeared anyway.


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[Assistant] Uh, once you have all the pi's, if anybody tells you, you know, how do they, how, how, if a hedge fund tells you these are the, these are the,


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[Assistant] These are the, you know, this is the revenue I'm going to produce for you in the next five years. If they tell you, if a company says, this is our business plan, we're going to put, we're going to build a factory today that's going to cost a certain amount of money, and we're going to get profits in the next five years, blah, blah, blah, blah, blah.


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[Assistant] You take the cash flows and multiply them by the pi's. So the correct price, P, is just whatever the cash flows you're predicting times these pi's that, uh, Fisher says is what you should really be finding.


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[Assistant] Nothing could be simpler if you know. Now why is this the right price? Because if you can go in the market and buy a dollar at time one for pi of one and a dollar at time two for pi of two, et cetera, you can buy all the cash flows from this investment project by spending this amount of money.


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[Assistant] So if the guy is offering you the investment opportunity at a higher price, it's crazy to do it. You could have bought those cash flows yourself by paying this price. If he offers it to you at a lower price than that, then definitely you should do it because you can, uhm, you know,


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[Assistant] It's a bargain, because if you had to buy it yourself, it would be more expensive. In fact, you can make an arbitrage profit. If he's offering it to you at a lower price, you can buy it. And how do you buy it? By selling these very promises, C1, C2, CT, on the market. And if people believe you that you'll pay, you can sell it for this price. So you buy his project for a lower price, you sell it.


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[Assistant] For a higher price, you make the difference, and when it comes time to keep your promises, the project is giving you the cash to keep your promises. You lock in a profit for sure.


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[Assistant] So if you knew the pi's, you would know for sure how to value, at any project where you knew for sure the cash flows, knowing the pi's would tell you how to value it, and would tell you whether it was attractive or not attractive. You just look at how high the present value is. Okay, any questions about that?


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[Assistant] So, we just figure out, we just need to figure out how to deduce what the little pi's are from the data that we're gonna be given. And that we are given every day by the market.


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[Assistant] A dollar tomorrow, and how much would it cost you? A dollar next year, how would you do that?


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[Assistant] You can trade, buy or sell in any of these treasury bonds. So you know, I, I'm, I, in the background I keep saying,


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[Assistant] That, uh, we're gonna have to worry about people defaulting. We're not quite doing it yet. So buying a treasury bond, you need the cash to buy it. Selling a treasury bond means you promise to deliver what the treasury bond promises, and you're selling it. You know, your promise is as good as the government's.


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[Assistant] So if you sell it to somebody, they'll pay you the same government price for it.


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[Assistant] So obviously in the background you're going to have to do something to convince the guy you're making the promise to that you're going to keep your promise.


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[Assistant] So we're going to worry about that later. So for now, when I say that the market for those treasury bonds


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[Assistant] Clear is that those levels, I mean that anybody who wants to can buy treasury bonds at those levels or can sell them even if he doesn't have them at those prices by making the promise of what the treasury bond does. Because we're assuming the government


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[Assistant] And you, everybody is just as reliable. Everybody's gonna keep their promise.


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[Assistant] So whether it's the treasury making the promise or you making the promise of the same thing. Okay, so how do you buy a dollar next year?


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[Assistant] What would you do in the market with the treasury bonds to get one dollar next year?


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[Assistant] Well, uhm, you're supposed to be telling me. What do you want to do? I want to know exactly what to do. You can look at those numbers. By the way, are you all following these? I may be. This is mysterious. There's a one-year


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[Assistant] The three-year treasury bond pays three dollars the end of the first year, three at the end of the second year, 103 at the end of the third year, etc.


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[Assistant] Okay, and the one year treasury bond happens to be selling for this price. The two year happens to be selling for that price. So what can I do with all these bonds to buy a dollar next year? Back, alright, go ahead. A two year bond. It's next year, one year from now.


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[Assistant] A step ahead of me, I'm saying, what do you do, nevermind how much does it cost, what do you do today to get


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[Assistant] A one dollar next year. What transaction can you make today?


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[Assistant] What purchase can you make today to get yourself one dollar next year?


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[Assistant] Buy a one year bond. Well, that'll give me a hundred one dollars next year. I want one dollar next year.


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[Assistant] Okay, well that's exactly the point. Okay, right, I take that fraction, that's what I want you to tell me. So.


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[Assistant] Is going to be 1 over 101 times the price of the 1-year bond, because the 1-year bond is paying, uhm,


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[Assistant] The one your bond is paying $101. So you take...


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[Assistant] So now we know pi one. Well, how would you buy one dollar at time at year two, in year two?


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[Assistant] So there's a way of directly buying one dollar in year two, once you know what the, uhm,


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[Assistant] How the treasuries trade. So what I'm doing is I'm explaining the idea of replication


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[Assistant] Replication, uh, pricing, that's giving me pricing.


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[Assistant] And it's going to lead to arbitrage. They're all basically going, some more ideas here.


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[Assistant] Okay, so what I want to do is directly buy one dollar in year two. So I could probably go to a bank and they would actually


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[Assistant] See how they're gonna go out and have to buy the dollar for me. Uh, so they're gonna go out and go to the treasury market. And what are they gonna do in the treasury market to come up with my dollar in year two? They're gonna replicate


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[Assistant] My request for one dollar in year two. So what is this bank going to do in the treasury market?


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[Assistant] What does it have to do to get one dollar for sure in year two and nothing else?


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[Assistant] The two-year bond, you're not selling, you're talking about the two-year bond. So how much of the two-year bond are they going to buy? 110 seconds. That's right, that's very good. Why is that?


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[Assistant] In year two, we're talking about year two now, year two, the two year bond pays a hundred two. You get one over a hundred two of those, you've got one dollar in year two. So that's pi of two.


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[Assistant] Okay, so which happens to be 100.2. That's how much that costs.


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[Assistant] Okay, but is that all you need to do? I mean, is that it? Are you paying the right amount? Are you paying too much? Or what are you doing?


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[Assistant] You gotta do more than just that. Why is that? Yeah.


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[Assistant] Close, but not quite. Okay, so, so do your reasoning before you got to, you told me what else to do, which you slightly miss-said. So why do you have to do anything at all? Tell me the reason why you want to do something else. You're right, you're on the right track, you just


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[Assistant] Slipped up a little bit, so what do you, why do you, why not just stop here?


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[Assistant] Exactly. So that's exactly right. By buying the two-year treasury, you got 102 in period, you know, you got, by buying this fraction of the two-year, you got just what you want in period, uh, two, but you also purchased the coupon in period one, which you don't need.


[User] Please transcribe this text

[Assistant] So that's giving you more than you needed to buy. You've bought extra. So you're going to actually be able to, uh,


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[Assistant] The real, the cost of getting the dollar at the end of year two is a little bit less than what we've written so far because you bought more. So far this is buying too much. You bought the dollar in year two, you also bought a little bit in year one. You can now sell off the extra stuff you've gotten in year one to reduce your cost of buying that. So what should you do in year one?


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[Assistant] It's exactly what you were thinking, you just didn't quite say it, right? So what did you do in year one?


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[Assistant] At time one now, it's that number. So I can, so I'm getting, so is that correct what I've written here? That's what you said, that's not quite right, yeaah.


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[Assistant] Okay, so this is what he meant to say.


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[Assistant] So that's fine, you know, so okay, so that's what you do. Exactly, so everybody's following, you agree with me now, right?


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[Assistant] Okay, so does everybody see what's going on here? To buy a dollar at time two, you buy


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[Assistant] Uh, you don't get the whole two-year treasury, you buy one one hundredth second of the two-year treasury, so it costs you that amount of money, but that gives you a little bit of extra at time one. How much extra does it give you? Well, you've got two dollars extra for every


[User] Please transcribe this text

[Assistant] To your treasury, but you didn't buy a whole to your treasury, you bought that fraction of it. So it gives you this much extra, which you now get to sell off. So you're gonna sell it off for this price, pi 1. And of course we can plug in for pi 1 by putting 101 down here and putting 100.1 up here.


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[Assistant] Uhm, any questions about that? So that's gonna be some number, which I calculated, uh, again, which happens to be 96.2, 0.962.


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[Assistant] So notice, of course, it's getting cheaper to buy, you know, how much does it cost to buy a dollar in year one? To buy a dollar in year two is less. Okay, now what about pi three? How would we get pi three?


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[Assistant] We'll stop at pi 3. How would we get pi 3? And then we're going to find a very fast way of computing all these numbers. What's pi 3? How would you get that? Yup.


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[Assistant] So the three-year bond costs 100.3, but we don't need all of it. We just need


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[Assistant] Right, because we got this extra stuff that we didn't need.


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[Assistant] Okay, so he's saying minus 3 times 103 of pi of 1. Okay, because we didn't need that. So is that the right answer?


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[Assistant] Not the right answer, but it's close. So what did he overlook? So he said, you buy the three-year bond,


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[Assistant] So by buying the three-year bond, you're getting, if you bought the whole three-year bond, you get three, three, a hundred, three. You only want one at the end. You have to divide by a hundred, three. Now we get three over one oh three, three over one oh three, one. And so he's saying, we've got two extra payments, let's, uhm, sell them off.


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[Assistant] And so he sold, okay, so he sold them off like that. Actually, that's correct. So he sold them off, and so this one he sold off at Pi 2, and this one he sold off at Pi 1. So he's making use of the fact that we've already found out this price and this price. But actually, that's slightly...


[User] Please transcribe this text

[Assistant] Okay, but what would you have to do? So we're talking about not what you would do, talking to the banker. We're talking about what the banker would do, and he's got a trade in the treasury market. So how's this guy gonna do this? The banker in the treasury market now, this pie over two dollars, he's gonna have to hold this complicated portfolio. He's gonna, what's he gonna do? To, to, to, uh,


[User] Please transcribe this text

[Assistant] Okay, he's going to have to combine the one and the two year to do this thing and then the one year to undo that thing. So it's actually going to be, so in terms of trading


[User] Please transcribe this text

[Assistant] If you just had to trade treasuries, what would you do? So this is the correct formula. That's correct and we can figure out what that is. Okay, and so the correct formula is 91.68, 91.9917. Okay, but you see what you've done is


[User] Please transcribe this text

[Assistant] These are the kind of fictitious things that Irving Fisher has filed us to do. What you're really doing in the market is trading the treasuries. So here, you've traded a treasury. You bought 1 over 103 units of a three-year treasury. Now what else should you do? You've got to trade treasuries. How can you sell off this amount of money in year two? You have to sell some treasuries to do that. So what would you sell?


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[Assistant] The two-year coupon bond, and so how much of that would you sell?


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[Assistant] Well, this is the amount of money you have to, uh, get.


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[Assistant] The two-year coupon bond delivers how much money.


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[Assistant] So if you did 1 over 100 seconds, you would get 1, so you have to divide this by 102. Okay, and that you multiply by the price.


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[Assistant] By the price, which is .96, by the price, which is 100.2. Okay, so there's that term. Right? So what have we done here? We've had to


[User] Please transcribe this text

[Assistant] Sell off this amount of money. So how can you sell off this amount of money?


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[Assistant] Well, by selling one over a hundred second two-year treasuries, you're selling off a dollar, but you don't want to sell off a dollar, you want to sell off something smaller than a dollar, so it's that amount. You're selling that amount of two-year treasuries. But now, now what do you have to do?


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[Assistant] Now you see, you've bought some one-year dollars by getting the three-year coupon bond and you've sold off some...


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[Assistant] But you, but by selling the two-year coupon bond, you've made some promises in year one. So you've gotta net out all those things and do the right thing on the one-year coupon bond, right? So that looks a little complicated.


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[Assistant] But you can obviously do it by algebra. So everybody following?


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[Assistant] You're not following what the right thing to do is, but let's just say in words what we've done.


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[Assistant] In words, what we've done, as we've said, there are things you can actually trade on the market. Those are the treasuries. Those are our benchmark securities. Let's call them benchmarks.


[User] Please transcribe this text

[Assistant] Is some other maybe fictitious securities or new securities. The price of the zeros. Those are the basic building blocks that will help us evaluate the present value of any investment.


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[Assistant] So the reason why we know these prices is because we can replicate them by trading only the benchmarks, only the treasuries. So to get the one year zero, we just buy the correct fraction of one year treasuries.


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[Assistant] To get the two-year zero, we have to buy the correct fraction of two-year treasuries and sell the correct fraction of one-year treasuries. And that gets us that thing. So we've replicated the two-year zero by a portfolio consisting of being long for two-year treasury and short the one-year treasury. Right?


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[Assistant] To get the three year zero coupon, we have to buy the three year treasury.


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[Assistant] Sell the two-year treasury and do something complicated that we haven't quite figured out with yet with the one-year treasury. And that will duplicate the three-year zero.


[User] Please transcribe this text

[Assistant] And then we'll just add up the cost of that portfolio that replicates this and that must be the price of that thing. Okay, so that's what we're doing. Are there any questions about that? Are you following this or? Yep.


[User] Please transcribe this text

[Assistant] So pi of one is what you pay at time zero to get a dollar at time one.


[User] Please transcribe this text

[Assistant] Pi of two is what you pay today at time zero to get a dollar at time two. Okay, so knowing those little pis, you can evaluate the, the


[User] Please transcribe this text

[Assistant] Price of anything just by multiplying the little pi's by the cash flows in the future.


[User] Please transcribe this text

[Assistant] And now the trick, this is a trick we're gonna see over and over and over again, the subtlety in finance is that they don't just tell you what the little pies are, you have to figure that out yourself.


[User] Please transcribe this text

[Assistant] Okay, and so how are you going to figure out the little pie? Well, you know that you know the treasuries. You can trade the treasuries. You know what those prices are. You can see it on the market. So by combining the treasuries in a very clever way, you can end up getting the prices


[User] Please transcribe this text

[Assistant] Of all the, the zero coupon bonds, the things that pay just a dollar at the end. Why are they called zero coupon bonds? Because it's like you just get principal at the end of one dollar without any coupon in the middle.


[User] Please transcribe this text

[Assistant] So the, the, the little pi's are called the zero coupon prices because the payments are just one dollar, you know, pi three is the price of one dollar at time three. It's as if there was a bond that paid no coupons and paid a dollar of principal at time three. So the little pi's are the prices of zero coupon bonds of various maturities.


[User] Please transcribe this text

[Assistant] And those aren't really traded directly in the market. What's traded directly in the market where, you know, pieces of paper change hands are the treasury bond. But everybody


[User] Please transcribe this text

[Assistant] Every day is calculating these zero coupon prices because that's what they need to do to evaluate


[User] Please transcribe this text

[Assistant] Okay, so this formula is going to be slightly complicated. I don't know whether it's worth writing down, so we've got


[User] Please transcribe this text

[Assistant] Buying the three year treasury, the right amount of that, then we have to sell a certain amount of the two year treasury. Okay, because we accumulated extra coupons. But now,


[User] Please transcribe this text

[Assistant] We're also going to be able to sell a certain amount of the one-year treasuries. And so how much is that going to be? It's going to be some formula, okay? So it's going to keep track of everything we did and get a formula here.


[User] Please transcribe this text

[Assistant] So I'm actually not going to bother, I think, I was going to write down the formula, but it'll take three minutes to work it out, because there's a much faster way of getting all these numbers.


[User] Please transcribe this text

[Assistant] Is everybody with me here? You all understand how I could get this number if I wanted to do the work together. If I figure out I had to sell, you know,


[User] Please transcribe this text

[Assistant] Sorry, I would do something with the one-year treasury here to compensate for the fact that the three-year thing I bought is paying me coupons here, the two-year thing I sold is reducing some of those coupons, and so it's only the net coupon that I can sell, and I'm gonna sell that by selling the one-year treasury.


[User] Please transcribe this text

[Assistant] Okay, so that's how I would get the number there, and I added the cost of doing all those things together, and I get .917. So, uhm,


[User] Please transcribe this text

[Assistant] So, it's too complicated to just figure this stuff out all the time. Alright? So, instead, there's a very fast algorithm.


[User] Please transcribe this text

[Assistant] That you can do almost instantly, and that's why it's such a triviality to calculate these numbers every day.


[User] Please transcribe this text

[Assistant] So it's called the principle, it's the principle of duality. You go backwards, and you say to yourself, you know,


[User] Please transcribe this text

[Assistant] Uhm, what I want is pi 1, pi 2, pi 3, pi 4, and pi 5.


[User] Please transcribe this text

[Assistant] And I've started to figure out what the replicating, so these are the prices of zero. Prices of zero coupon bond. Zero coupon bond. That's what I want. Want prices of zero coupon bond. I have the prices of the treasury.


[User] Please transcribe this text

[Assistant] And the way I'm figuring out the prices of the zero coupon bond is by replication. Now, if somebody, stupidly, as happened 50 and 60 years ago, fairly routinely, if somebody was willing to trade me, you know, to, to, someone was willing to give me a dollar,


[User] Please transcribe this text

[Assistant] Uh, in year three, and only asked 90 cents for it.