econ-251: Financial Theory
Lecture 2 - Utilities, Endowments, and Equilibrium [September 8, 2009]
Chapter 1. Introduction [00:00:00]
Professor John Geanakoplos: So for those of you who weren't here yesterday--or, last class, first class, I'll say a couple words about what happened, basically four words. The course is really made of up four different elements. The first part is the standard financial theory course that grew up in the last ten years at a lot of major universities, pioneered by a bunch of guys who won Nobel Prizes in business schools. And it's the method, some of them quite clever and I think fun, methods for pricing financial assets and making optimal financial decisions. So you're going to learn all these tricks and how the financial system works, and you'll learn it both from a theoretical point of view, the way they thought of it in these finance schools, and also from a practical point of view since many of these very same problems come up all the time in the hedge fund I helped start.
So that'll be the main part of the course, but there are three other things that I want to concentrate on in the course. So the second point is reexamining the logic of laissez-faire and regulation. This is a dramatic moment in our history now where there's tremendous pressure on--temporarily anyway--on the government to establish all sorts of new regulations.
There's also tremendous resistance to establishing the new sorts of regulations. So there's a debate going on now in Congress and in the halls of academia about what kind of regulations should we put in place, what regulations would have prevented the crisis we've just lived through. The crisis, by the way, which I don't think we're done with yet. So there's a very powerful argument in economics. The most famous argument in economics, the invisible hand argument that basically says markets work best when they're not encumbered by government interventions. So we're going to reexamine that argument in the context of financial markets.
Then the third thing I'm going to discuss in this course at some length is the mortgage market and the recent crisis. After all, my hedge fund is a mortgage hedge fund. We founded it in 1994, by the way, which was--the five years before that I was running the Fixed Income Research Department at Kidder Peabody, which was the biggest player in the mortgage market then on the sell side. The hedge funds buy mortgages, investment banks create and sell the mortgage securities. So I was running the research department at the firm that did twenty percent of the market in what's called CMOs, and then I changed to the buy side and was at a hedge fund that bought those kinds of CMOs, and bought sub-prime mortgage, CDOs, everything. So it seems I suffered greatly through the last two years of the mortgage crisis and it would just be foolish not to explain what was going on and what it felt like to be in a mortgage hedge fund while the rest of the world was collapsing around us. And for quite a while it's given me some great embarrassment to have been part of it all.
On the other hand, now I feel like one of those survivors. Hundreds of our counter-parties and much bigger mortgage players went out of business and we didn't, so I don't feel quite as bad about it as I did before. And I don't know every detail of what went on in my hedge fund, because after all I'm only part time there, but there's a lot I do know about and so I'll try and tell you some of that.
And then the fourth thing is Social Security. This is the biggest government program and it's a financial problem what to do about retirement, and Social Security is the biggest government program of all. The only thing close is the military budget in terms of annual expenditures. And so I'm going to explain how that works, and what the problem is, and how it arose, and what I think the solution is. So those are the four things the course is going to concentrate on.
The mechanics of the course, again, are homeworks, every week there's going to be a homework with little problems illustrating what we're talking about. So there's one already on the web, Tuesday, today, every Tuesday there will be one on the web. It'll be due the next Tuesday. The sections will always meet between Thursday and Monday, so the problem set will come Tuesday. You'll have two days of classes on the material that the problem set will cover and then you can talk to the TAs about stuff between Thursday and Tuesday when I presume you'll do the problem set. And that's twenty percent of the grade. Twenty percent is the first midterm. Twenty percent is the second midterm and forty percent is the final.
Two midterms takes a lot of class time, on the other hand, and it also takes a tremendous amount of effort by the TAs. And so I appreciate their willingness to grade two midterms, but I think you'll find it's helpful to study the course in two pieces than try to do the whole thing--It'll be much better for you I found in the past to have two midterms.
Oh, and one warning. The course doesn't require difficult mathematics, but for me, as I said in the first class, it's very interesting that there are so many subtle things that affect a financial decision, and you have to think about what you know and all the different things you know. You have to think about what the other guy knows who's taking the other side of the market. You have to think about what he knows about what you know, and you have to think about what he knows about what you know about what he knows, and all that thing in the end boils down to one number, the price. So it's a philosophically interesting problem, interactive epistemology. Some people have described economics as interactive epistemology.
It's more complicated than standard epistemology and philosophy because there they go in circles thinking about what one person knows and whether you can know that you know and stuff like that. In economics you have to worry about what you know, what the other guy knows, what you know about what he knows about what you know etcetera. So it's a more complex problem, and yet at the end there's just one number which can be right or wrong.
And so when I was a freshman here at Yale my roommate who was a classics major said that his subject was much harder than mine--that was math--because all I had to do was be right. And so I'm going to take advantage of that simplicity and every problem is going to have a number that you're supposed to find. And so it's not complicated mathematics, but it involves lots of numbers. And so if you hate numbers you shouldn't take this course. And as I said before, there have always been people who, you know, you can be very smart. You can also have a great future in finance and not like numbers. You can like making deals and things like that not thinking in terms of numbers. So just because you don't like numbers and maybe shouldn't take this course doesn't mean you should be discouraged about finance. It's just how I happen to teach the course because that's what's comfortable for me. So I'm just warning you about it. It won't be hard, but it'll be relentless.
Chapter 2. Why Model? [00:07:04]
I want to talk today about that second problem, about the logic of the free market and to do that I'm going to have to introduce a model. So it raises the question of what is a model in economics. Many of you have taken economics before. You sort of know what this idea is, but I think it's worth spending a minute on it because it represented a revolution in thought. So for an economist a model means you distinguish exogenous variables from endogenous variables. The exogenous things people can't control. They're just the weather and things like that. The endogenous variables are things they can control and you're trying to predict what the endogenous variables are going to turn out to be—like, what will the prices be, what will the consumptions be, things like that. How much income will people have? Those are the endogenous variables.
So you have a theory. So the theory is couched in terms of equilibrium. There's a bunch of equations which have to be satisfied, F of E and X. So given the endogenous variables and the exogenous variables, exogenous and endogenous, I wrote them in that order, there's a set of simultaneous equations, F, that have to be satisfied. And so you find equilibrium when given the exogenous variables E you find the endogenous variables X of E that solve that system of simultaneous equations.
All our equilibrium models are going to have that form, and one very important thing they allow you to do, which is the heart of economic analysis is comparative statics. If you change the exogenous variable E it'll require a different X to solve the equation. So E has an effect on X in order to restore equilibrium. And so the prediction that a change in E has a certain effect on X is called comparative statics. Now, how would a historian describe that? A historian would say, "Well, that's counterfactual reasoning. The environment is E. Why are you bothering to tell me about what would happen if the environment changed from E to E-prime?"
Well, that's the heart of economic analysis. So in history you hardly ever get much. People talk about it a little just to raise the question. How would the Vietnam War have gone if Kennedy hadn't been assassinated? So they all bring that up, but you get two sentences. "Oh, he was really going to pull out," or "Oh, he had been sending more advisors. It would have gone the same way." That's about it. In economics the heart of the thing is to go off on a tangent and figure out what would have happened if the environment had been different.
So why do a model? Well, because many different settings can be described by the same model. So it just saves time. It makes things much simpler. From the counterfactual reasoning you're making predictions. It helps your understanding. And then for the purposes of the next few lectures the most important thing is there's some properties of equilibrium. Like, for example, equilibrium is so good you wouldn't want to interfere with equilibrium because it makes everyone so well off and it would be a terrible thing to regulate. So those properties of equilibrium are what we have to test the logic of.
So there's an obvious critique you can make of modeling. The first person to make a model was Ricardo, who you I'm sure have heard of, the principle of comparative advantage. He was the first guy who didn't write verbally. He said, "Okay, I'm talking about international trade and why free trade is a good idea. I could make a verbal argument. That's what everyone else has done, but I'm not going to do it. I'm going to say, 'Suppose that England produced with one hour of labor three bottles of wine' and so on, " and he had a little numerical example. And he solved it and he showed that in that numerical example it's better to have free trade as paradoxical as it might have sounded at the time. The Portuguese had such lower labor costs why shouldn't English workers be afraid of being thrown out of their jobs when trading with Portugal where the labor was so much less expensive.
Well he explained why that turned out not to be the case, but in terms of a model. So Malthus, who you've also heard of, a contemporary of his and his rival but also his friend, said, "This model stuff is ridiculous because if you start making a model the point of a model is to make deductions from it, and to analyze it, and analyze it deeper, and deeper, and deeper, and of course the model to begin with is going to be wrong and as you go deeper and deeper into the analysis of the model the error that you made at the beginning is going to get compounded." Like he said, "The tailors of Laputa, who by a slight mistake at the outset"--doing their stitches, go wrong--the stitch goes further and further wrong--you "arrive at conclusions the most distance from the truth."
Anyway, that's what you might think is wrong with models, and the very first model was critiqued by that reasoning, but it's turned out historically that modeling is the way to make progress in economics and everybody does modeling now. You'll find out as I talk more about it that the Cowles Foundation, which has been at Yale since 1955, was founded by a Cowles [correction: Yale] undergraduate. You'll hear the whole history of it. I was the director of it for nine years. That was the one most important institution in the world promoting the uses of mathematics in economics, and the revolution succeeded and now all economists use models and mathematics. Anyway, let's take an example of the simplest model.
There are so many different ways of organizing price and trade. At a supermarket the seller just sets the price and you decide to buy it. If you go to Jerusalem or something in the old city you know that you're haggling over everything. You offer this and the guy says no and you walk away, and you come back and it takes a half a day to argue the price, but that's another way of arguing the price. Then there's--the government could set the price.
In the Paris Bourse the way it worked is there would be a temporary price set and then supply and demand--people would announce how much they wanted to buy, and if the supply didn't equal demand the price would get changed. So it was--tâtonnement means groping, or groping to the price. There's the commodities futures just like the experiment we ran where people yell at each other. There's computer bid/ask prices where you do everything online. There's the specialist in the stock exchange. There's one guy everybody has to come to, and so he's responsible for clearing the markets. So I might, in fact, mention a little bit of the history of this sort of thing. I don't know if I can hit escape. It might be somewhat interesting.
Chapter 3. History of Markets [00:13:30]
So the first people who had these well-developed markets and money were the Lydians. They invented money in 640 B.C. and they had gold coins, and with all this money and trading they got very quickly to gambling and prostitution for money. And Midas, the Midas touch was everything turned to gold was Lydian. They've discovered all these mints where their capital was so they know that they were making all this money and gold and stuff like that. So they had open-air markets. They invented the retail markets.
Croesus was one of the most famous Lydian kings, and he's the guy--rich as Croesus is a famous expression. He's the one who went to the Delphic oracle and asked if he should fight the great Persian, Cyrus the Great and the Delphic oracle as usual mysteriously said a great kingdom will be destroyed, and since it was Cyrus the Great he figured it must be Cyrus’ and it was his kingdom that was destroyed. So the Greeks copied a lot of that stuff. They had their agora which was the open market and they had lots of trade, and they understood supply and demand, by the way. This isn't the modern example. In the politics there's a story of Thales who predicts a bad harvest. He's a great mathematician and astronomer, and he predicts a bad harvest and he figures if he corners the wheat market he'll make a fortune, which he does.
Aristotle was famous for saying, "Money is just a convention. It's not really worth anything. People just agree it's worth something even if it's just pieces of paper or coins that worth much more than the coins and how could that be," and anyway there's a long political connection to that, the difference between nature and convention, but anyhow he also said, "Loaning at interest was unnatural and terrible," but all the while he was saying it the Delphic oracle was lending at interest. Economics is a Greek word, household management, Xenophon wrote a whole book about it. And just one more little history or historical thing, Hermes, the messenger god, the god of information, so remember the modern financial view of information, markets and information, anyway he was the Greek god, messenger god, and god of information. The word commerce comes from Hermes. And the Romans who took over the same god and called him Mercury that's where we get the word merchant from and market.
Anyway, all right I'm not going to bother with all this. I used to go on and on about this. So the point is that in ancient times the market was already established and this idea of supply and demand had already been created but there are many different kinds of markets, as I've just said, and they work in many different ways, but we're going to describe them in one model. So just to mention a couple of others, the model, the experiment we ran in the class last time is called the double auction, and the experiment I told you about and had you do was actually an experiment that has been run before. And for the last ten or twenty years many economists have run these sorts of experiments.
It's amazing that before that, before twenty or thirty years ago no one thought to do that. You didn't think that students with no training and no experience could ever be led to do something that was sensible, but actually you did quite brilliantly. And by the way, I've been told that those of you who performed, maybe you're still in the first two rows, you have to sign, even those of you who were left at the end unable to trade you have to sign a release form so you can't sue Yale for your disappointed faces appearing on the internet afterwards.
So anyway, the fact is we're going to see that the people who were left at the end were actually very rational. In fact nobody made a mistake. I've done this experiment now ten or twenty times and I would say that half the time somebody buys something for more than it's worth to them. Nobody made a mistake and it almost came out exactly as it should, but we'll come to that in a second. Anyway, that double auction is the most complicated kind of auction, but auctions have been run for a long time.
The first recorded auction you may have heard about was--Herodotus describes the Babylonian auction in 500 B.C. These are all going to be very politically incorrect, but a lot of economics is politically incorrect. Anyway, the first auction in 500 B.C. was the Babylonians auctioned off all the 18-year-old women as wives and they auctioned them in order of most beautiful to least beautiful. And so they got a very high price and the price went down and down and down until it hit zero and then it started going negative, but they used the revenue from the first wives to subsidize the husbands who would accept the other wives as it kept going down.
The next most awful auction was the Roman Empire itself was auctioned off. So if you saw the movie Gladiator you may remember that Marcus Aurelius is the great emperor, and he dies, and then the evil Commodus takes over, and he dies as a gladiator there. And then there's the senator who you sort of hardly ever see, but you know he's the good senator who somehow--he appears a few times--you know that he's a good guy and he's going to take over. So his name is
Pertinax, and he does take over. But he's a good guy and he gets killed almost immediately by the Praetorian Guard and the Praetorian Guard then doesn't know who to make emperor so they auction the whole empire off. And so it's bought by Didius Julianus, and he doesn't last very long, and he gets killed too. The Roman legions come back and kill him.
So anyway, I grew up in Urbana, Illinois and I used to go to these livestock auctions where they'd sell something. They'd talk incredibly fast <<speaking gibberish very fast like an auctioneer>>. They talk like that and I don't know how anybody can understand them, and then there's the famous slave auction, so--where they'd actually auction slaves, and you've seen it in the movies maybe. And that's where the expression, "Going once, going twice, third and last call, going, going, gone," that's what they used to say at the slave auction.
So the double auction that we saw was kind of what happened at the beginnings of the New York Stock Exchange. The first traded securities--there were only five of them, so how did they start? There was the Revolutionary War. A lot of states had borrowed money and issued their bonds, and there are two banks, Bank of New York and the National Bank of the U.S. that had issued bonds. Those were the only tradable securities. And so a bunch of states had issued bonds. So what happened was after the Revolutionary War most people expected the bonds wouldn't be paid back. After all, there was a huge expense fighting the Revolutionary War. The government didn't have very much money. The price of the bonds had already collapsed, and Jefferson wanted the U.S. to just, you know, wanted to leave the states and let them default.
And Hamilton said that that would be terrible, that the reputation of the country was going to be made at what happened at the very founding of the country and it was important that the U.S. never break a debt. So he persuaded Washington to have the federal government buy all the debt of the states and basically pay it all off, so none of the debts were broken. Jefferson argued, "That's crazy. The people who originally bought the bonds, who lent the money to the government, the farmers who did it they didn't own the bonds any more. They probably all sold it for twenty dollars. It was all this despicable speculators who held the bonds. You're only going to enrich them by paying them off." So he just wouldn't budge.
And finally Hamilton, supposedly--this is a famous story, I assume it's true--Hamilton went to Washington and said, "All right, move the capitol from New York to Washington. That'll make Jefferson happy because it's near his dear Virginia and in exchange get him to concede that we have to pay off the debt." So Washington brokered that deal and the debt was paid, and the U.S. since then has never defaulted on its debt and virtually no other country can say that.
For example, Russia has never paid a thirty-year debt. It always has defaulted, and we'll come back to that a little later when we talk about the crisis of '97-'98. Anyway, so these five securities--three government bonds and these two from the Revolutionary War and two banks--were the only securities sold and they used to be sold every day in a double auction exactly as the kind that we described where people would yell and scream at each other and the whole thing would be over in a few minutes, and that would be it for the day, and then the next day they would do the same thing over and over again.
Well, they had to stop that when Alexander Duer, who was Hamilton's assistant, started using his inside information about whether the government was or wasn't going to make all its payments and whether they're going to issue new bonds and stuff like that to try and speculate on the market. And he would do it all by borrowing. He'd borrow a huge amount of money and with the borrowed money he'd buy bonds, and if the price went against him he'd lose a lot more because he was leveraged. And so it caused gigantic gyrations in the market and the whole thing had to be changed, and it was made a much smaller group of people.
Anyway, so that was the beginnings of it. And we're going to come back to that because that view of the gyrations of the market being caused by too much borrowing and speculation is exactly the view that I'm going to take in explaining the most recent crisis. So anyway, you remember what we did in our experiment. We had eight buyers whose reservation prices are those eight numbers up there. That's what each person thought it was worth to him. Each person knew his own price, but not any of the others. I told you almost nothing about what was going on. There was some context. I gave an example of a person who thought it was worth fifteen, so you had some idea, probably, from that example that the numbers weren't ten thousand, plus you knew your own number.
But other than that you knew absolutely nothing and each buyer knew her own number and not any of the other numbers. So here we have sixteen different pieces of information. Everybody has an incentive to keep her information secret. Why should anybody admit that she's willing to sell at six? She'll get a worse price. She's going to lie and say the thing is much more. She's going to make an argument that says, "Well, these are football," okay, I better try the guy here. The forty-four guy, he's going to say, "This is a football ticket." No, sorry. What am I going to do? Let's say she's a forty-four. She's going to say, "Football tickets, they're completely worthless." I'm doing a stereotype. "These are completely worthless. Who would want to go to a football game? I certainly don't want to go to a football game. They can't be worth any more than twelve or something." So all the buyers, the blue buyers, are going to be making arguments suggesting the price should be low, reasons why the stuff really isn't worth very much. All the sellers are going to be making arguments saying the stuff is intrinsically incredibly valuable. Football tickets are incredibly important.
Chapter 4. Supply and Demand and General Equilibrium [00:24:41]
That's the facts, now you need a model and a theory that fits the facts, and I'm belaboring the obvious, but the obvious is always central to everything, the obvious theory would go something like, well, somehow these people are going to get matched up and maybe thirty-eight will sell to forty-four and all eight things will be sold. And the more transactions you have the better. And what else might a theory say, a wrong theory? It might say the more people in red, or the more people making arguments that the price should be higher the more compelling the argument will be. You'll be overwhelmed by numbers and you'll think that the price should be higher because more people will be arguing for a higher price.
But the theory, the economic theory is the exact opposite of all that. So the economic theory is quite a shocking theory, I think. It starts with a situation where people are arguing and talking about the price. They're not doing anything else but making arguments about the price and making offers about the price. They're haggling about the price. The whole of the activity is about the price and how to change it and what it should be.
The economic theory, the first theory, the most important theory of economics, supply and demand, is that--so that describes what happened, is the exact opposite. The theory says let's suppose that a price appeared out of thin air. There was no arguing about the price. Nobody even thinks they have any chance of changing the price. Somehow a price gets into everybody's head, the price of twenty-five and at that price of twenty-five everybody who wants to buy buys as much as they want. So mister forty-four he thinks the ticket is worth forty-four. If he can buy it for twenty-five he'll want to buy. Forty thinks it's worth forty and the price is only twenty-five so, again, he's going to gain by buying, he'll want to buy. Twelve thinks it's only worth twelve. He's not going to pay twenty-five for it. And similarly the sellers, seller number ten, she's going to say, "Okay, I can get twenty-five for it. It was worth ten. It's a good deal for me to do."
So the theory says somehow miraculously the price comes out of thin air. It's given. Everybody taking that price as given, figuring they have no power to change it, buys or sells all they want at that price. And so that's the theory. So it's price taking, out of thin air. The price comes from somewhere. Everybody acts by maximizing, doing the best for them given the price. They all understand what the price is, and the price has miraculously been imagined at exactly the level that will clear all the markets. So everyone who wants to buy is able to, and everyone who wants to sell is able to.
That's the theory. The theory's completely the opposite of what common sense suggests since, as I said, the whole thing was this grappling and groping and pushing and shoving and yelling and arguing about what the price should be and the theory says nobody says a word about the price. They just take it as given and then they act after that.
So the most basic economic model is a paradox, and good economics is almost always a paradox. If you want to make a convincing economic argument you almost always say it in a paradoxical way. And so going back to the very beginning where we said what a model is, the standard economic model is you take the exogenous things, which in this case are the reservation prices of all the people, you have to solve equations which are here, supply equals demand, which determines the endogenous variables, which are the price and who buys and who sells. And the reason the theory is always often paradoxical is if you change some exogenous variable it looks like it's going to move things in a commonsensical direction, but then when people react to the changed environment--X is a reaction to the change in E--and the change in X might be so big and so important that it reverses the apparent change in E.
So you get these surprising conclusions. "If everybody tries to save more," Keynes said, "It may be that everyone will end up saving less," things like that. So economics at its best takes advantage of its paradoxical nature at its heart and uses that as a rhetorical device. So it's a non-obvious theory.
Now, why do we believe the theory? Well, all those different examples I gave you of markets they all seem to fit. I forgot where they were and I don't even remember what they were. I don't remember what they were. The shopping center thing, the haggling, the tâtonnement Bourse, the commodities futures, all that, if you look after the fact at what people wanted to do and what price emerged it seems to fit the theory.
So there's overwhelming evidence that this theory seems to work. And you saw that in our own example, in our experiment where you had no training at all, it came pretty close. So all these five red sellers they all sold, I think, and the five buyers the only difference was that instead of twenty-six buying twenty bought, and the prices were all between twenty and twenty-five, so they weren't exactly twenty-five, but they were very close to twenty-five. And the ten people who were supposed to have bought and sold, well nine out of the ten actually did buy and sell. So it's pretty hard to match a theory like that with so little practice. I mean, I've always found it quite astonishing.
Why is this happening? Does anyone want to make a comment or ask a question about this theory? All right, well what are the properties of equilibrium you get out of this? Well, everyone trades at one price. So this is going to be very important for finance, the idea that there's one price for everything. Then you can also define the--so you know what the theory is. I already told you the exogenous variables are the reservation values. The endogenous variable is the price that emerges and who buys and who sells. So why is this such a good outcome?
It seems like a terrible outcome. There are those six people standing there at the end unable to trade, facing the camera, looking slightly embarrassed that all their friends managed to buy and sell and they couldn't do it and what's the matter with them. So they feel bad. They feel discriminated against. It doesn't look like it's such a great thing. We know that there's another way of making all eight buyers purchase from all eight sellers just by doing the corresponding one above. What's so good about the market outcome? It actually doesn't sound so great. Well, the answer is it is great and what's great about is that within two minutes the market figured out enough about what everybody valued the football tickets at to put the football tickets in the ten peoples' hands who valued them most.
All right, so in the end those five blue guys--almost without that one exception--and the one, two, three red sellers, those three sellers and those five buyers, the top eight people ended up with the eight football tickets and the bottom eight didn't end up with any football tickets. So the football tickets got put into the hands of the people who valued them the most. And so, as I said, if you just simply sat there and went through sixteen tickets and sorted them into most and least and then tried to arrange all the football tickets it would have taken almost as long, and that would have been with benefit of knowing what all the numbers are.
Here the market does it not knowing what the numbers are and the only accessed information is through people who don't want to reveal their numbers, and still the market figured it out. All right, so that's the message.
So we have a model which is surprising, which seems to describe the facts, and which gives us a surprising conclusion and an incredibly important conclusion. The market is an extremely useful mechanism of eliciting information and turning the information into something that allocates things efficiently, and you couldn't do better than that. No other arrangement would have put football tickets in the hands of people who like them better. So Hayek described the market as a great calculating machine, and well so it is. Now, there are a couple other things that you can get out of this model.
Another lesson of this model is that the equilibrium price is equal not to the average of the price of the buyers, or the average of the price of the sellers, or the average of all the prices or something like that. It's equal to what the marginal buyer thinks it's worth. So there's a critical marginal buyer and marginal seller. They're almost indifferent to buying or selling. They could go either way. They're pretty close to buying or selling. The price is going to turn out to be very close to that valuation of the marginal buyer.
So somehow the margin is going to play a big--so the word marginal, this is an invention in 1871, is going to play a big role in economic reasoning. So it gives us a completely different understanding. You might think that the price of tickets has something to do with their total value or average value or something like that. It's got to do with the value of a marginal person, the person just on the edge.
So then the comparative statics are that the, as I said, the surprising thing that if you change a non-marginal person, you take mister forty-four, the buyer at the top, you change him to fifty. Looks like the buyers are now more desperate to buy, won't have any effect on the price. You change that seller, miss six, you change her to two or to eight, again, it'll have no effect on the price, because those two people, the guy at forty-four and the lady at six, they're not marginal so they don't affect the price. You add some more buyers you might think that they're arguing for the price to be lower, as I said you're going to end up raising the price or else having no effect on it if they're not marginal.
Now one more thing, one last thing, one last message of this model, if you didn't know--we knew the reservation prices ourselves because I set up the experiment, but if you didn't know it you could infer something from the price. So part of finance is going in the backwards direction. The theory says take the exogenous variables. Predict what the equilibrium's going to be. Financial theory does that, but often it goes in the reverse direction. We can see what the prices are. That must tell us something about the exogenous valuations. So financial theory says, "Well if the price is such and such it must mean that at least the marginal person values it at such and such and so that's why the price is that. It's the value of some special persons."
So we'll come back to that argument. So that lesson of economics, that's the first economic model, the most important economic model, we're going to now have to generalize it in all kinds of ways, but it's always going to come back to that same message. And so Adam Smith he was the one who first invented the invisible hand. There was nothing mathematical in what he said. Ricardo was the first one to make a model. Marx said, I don't have time to talk about Marx, but he had quite elaborate models, actually, and his verbal arguments conceal a huge mathematical apparatus.
On his deathbed, by the way, he was trying to learn calculus, incidentally. So Jevons, Menger and Walras 1871 right after Marx's famous Kapital came out in 1867 they invented the idea of the margin and things like that and the critique therefore of Marx, and Marx was trying to figure out what they were all about. Anyway, Marshall was a great economist, Fisher, Samuelson, Hicks, Arrow, Debreu; these are the most famous people who extended this model and the logic of laissez faire and regulation which we're going to come to.
Now what are the two ways we have to generalize, there are three ways we have to generalize the model. We have to think of many commodities, not just one. We have to think of people buying more than one unit of a commodity. That's called general equilibrium. And then we have to put in financial things. We have to put in stocks and bonds and things like that. It sounds like things are going to get so complicated, but in fact it turns out I'm going to spend another class after this talking about this. There's not that much complication to get all those things in. There'll be two more classes about this.
So I'm recapitulating all that you have to know for the purposes of this class from introductory economics and intermediate economics. The only thing you have to know you'll hear now in these two classes and some of you will find it's incomprehensible, and so that's one good reason for doing it now. You find out right at the beginning whether it's too complicated to bother with.
Chapter 5. Marginal Utility [00:37:59]
So anyway, I'm going to keep going now to extend the model. So the biggest advance, the next advance, sort of, which was related to this is Adam Smith said, "How could it be that water which is so valuable has such a low price, and diamonds which are so useless, basically, to everybody has such a high price? I mean, there's not some marginal buyer who thinks that diamonds are somehow more important to him than water, so how could it be that water's got a much lower price than diamonds and everybody would say that it's more valuable?" Well, to answer that question what we have to do is we have to imagine that people are capable of consuming more than one good. So for instance, let's imagine that there's good X here which is the football tickets we had before, and you remember our numbers.
Let's just go back to the numbers for a second. I'll stay here for a while. The first buyer thought one ticket was worth forty-four. A second ticket was useless to that buyer. Well, suppose we write utility here. Now, this first buyer--let's put this forty-four here--this first buyer you might say got utility of forty-four for holding one ticket. If he held half a ticket maybe his utility would be twenty-two. Now, in fact we know that half a ticket doesn't get you into a game so his utility would really be zero. When we're talking about thousands of tickets to a football game a half or one it's not so important. Let's just say his utility went up linearly with the quantity of tickets he had. To make a discrete variable a continuous variable his utility goes up linear at the rate of forty-four per ticket. Well, after one ticket he gets no extra utility out of holding any more tickets so his utility might look something like that.
But now let's imagine he wanted two tickets and that the first ticket was important to him and the second ticket he could take his girlfriend, let's say, but he's not quite as worried about her as himself. So let's say that he, for the second ticket, gets an extra forty utils. So after you get to ticket number two his utility is going to be up to eighty-four, which is forty-four and forty. Now you notice that the rate of increase of utility per unit of ticket is forty-four here and then it switches to forty. Okay, now, why do I--why do I--okay, and if he wanted one more ticket maybe he'd only get utility of one-twenty for the last ticket. So for a third ticket his utility would--three goes up like that, utility would go up like this. It's a little flatter again. So here we have a utility function which is increasing the number of tickets you hold. It's not restricted to just having one ticket, but the rate of increase goes down as you get more and more tickets from the rate of increase of forty-four, to the rate of increase of forty, to the rate of increase of thirty-six.
Now, if you ask this person how many tickets does he want to buy, well what's he going to say? How's he going to figure out how much to buy? This is his utility, but now I claim this person buying multiple tickets is going to behave exactly like the top three people up there would have behaved. So his utility at the top for three tickets is one-twenty, for two is eighty-four, for one is forty-four.
Those sound like important numbers, his total utility, but actually they're not important numbers. The important number is the marginal utility. So the marginal utility, so if you go one, two and three here, the marginal utility for the first ticket was forty-four. The marginal utility for the second ticket was forty, and the marginal utility for the third ticket was thirty-six. So those are the important numbers, the same numbers that are up there. Why is that?
Well, let's ask the guy. This person who now likes three tickets, after here let's say he's flat so it goes down to zero, let's ask him how many tickets would he buy at the price of forty-two. Well, from this utility function you have to say if I bought one ticket I'd have a utility of forty-four minus--let's say my utility function now is U of X and money is this function of X. I'll call this U of X. I don't want to write it out. This is U of X plus M for money. So he says, "If I buy one ticket at a price of forty-four I lose forty-two from here, but I gain forty-four from here, so I probably should buy one ticket. If I buy a second ticket this number goes up to eighty-four and now this one goes down by forty-two twice, so maybe it's not such a great idea." So what is he actually thinking?
All he's doing is he's looking at the price in this axis and comparing it to his marginal utility, the extra utility out of getting an extra ticket. So if the price is forty-two here he's going to say, "Well, at a price of forty-two the first one's worthwhile. I'm getting more utility out of that. After that it's stupid to buy another ticket because I'm getting extra utility of forty compared to a price of forty-two." So he's going to do exactly the same thing as our single ticket buyers did over there. One guy whose utility goes from forty-four to eighty-four to one-twenty is going to behave exactly--provided he's got enough money to afford to buy at these going prices--his behavior will be exactly the same as the three separate individuals over there.
So in fact the marginal revolution--so Jevons, Menger, and Walras in 1871 all came up with the idea at the same time of diminishing margin utility, and they said if you have people who consume multiple amounts of every commodity but they have diminishing marginal utility they're going to behave very much the same way as this little example. So this little example, in fact, is going to be extremely instructive. In fact it contains all the kernels of truth of a more general model where people consume huge amounts of every good. Just that they have diminishing marginal utility. So I'm going to now describe a slightly more complicated--so I'm going to describe this more complicated model.
Chapter 6. Endowments and Equilibrium [00:45:20]
So what's the way of building a much more general, but hopefully still very simple abstract model of general equilibrium that will capture and generalize the example we already had? Well, the idea is to start with the exogenous variables--this isn't going to move so I don't want to do that--do this--the exogenous variables are going to be the people, so I'll have individuals, i in I, so let's call them individuals. So you see why I use the word I. i in I, and what is it that characterizes every individual, a utility function. So each individual is characterized by a utility and an endowment. So to start with let's say--so the individuals and we'll call the individuals and the goods c in C.
So let's just say there are two goods X and Y. So an individual's going to be characterized by utility function, it's a welfare function of X and Y equals ui of X plus vi of Y. And an endowment, Ei equals Ei of X and Ei of Y or (EiX,EiY). So for example you could have, I don't know, you could have, this could be--so let's just think about this. So this is exactly the kind of situation we had before. We had precisely this going on before.
What was the endowment? Every person began with money. It could have been money before and with football tickets. And we said that the story that--so these original marginalists argued that it's part of human nature that the more you get of something the less and less extra advantage it brings you. There may be exceptions. Maybe you need two of something. You need both shoes in order for the shoes to help, but every pair of shoes after that was going to be less and less valuable to you. And so beside from these small blips that come from indivisibilities or things like that peoples’ utility increases but at a smaller and smaller rate as they get more of everything. That's just human nature, they claim.
They even tried to measure utility. So they would try and measure the temperature of the skin and things like that and see how it increased when you gave people more of something and whether the rate of increase and how much they smiled and stuff like that whether that would actually change in a lesser and lesser way as you add more and more utility. Well, they abandoned that sort of thing eventually. But anyway, they kept the idea of diminishing marginal utility. So we want to keep the idea that ui of X and vi of Y show diminishing marginal utility.
So the way of saying that, I told you this is one of the--so the first handout in the reading list was review of mathematics you should know, or if you don't know you have to learn, diminishing marginal utility means something that looks like that. It's a concave function. So here's X. Here's utility, and here's ui of X, say. It goes up as you get more X, but at a rate that declines. So the slope is getting smaller and smaller. That's diminishing marginal utility. So this curve that's increasing, but a lesser and lesser rate we can approximate with a continuous differentiable curve that looks like that, so it doesn't have the kinks here, and that's exactly the kind of assumption that seems reasonable to fit the facts, and at least for consumption.
Our main interest, of course, is at the bottom here in financial equilibrium, but we have to know what's going on in the economy. All these finance professors, as I said in business schools, they ignored the part above. They started right away with the assets and the bonds. Said they didn't need to pay any attention to what was going on in the economy, because everything was going to be great. But we're going to find that there's a big interaction between the financial sector and the economic sector. That's going to be the heart of what we're doing even though it was ignored in finance most of the time.
So anyway, diminishing marginal utility for both of these, so for instance we could have a hundred X minus one half X squared plus Y. That's one example of a utility function. So that's going to be a standard kind of utility function. So the only two ones I'm ever going to use are things like this, or one-third log X plus two-thirds log Y. Whenever I write log I mean natural log. This is linear quadratic. So this is quadratic, in fact linear quadratic, so maybe both will be quadratic, and this is logarithmic.
Now both of these have this property of diminishing marginal utility because I can take derivative of this, the derivative of one hundred X minus one half X squared so the marginal utility of X is equal to one hundred minus X, and that obviously declines. So it's diminishing marginal utility. And then the derivative here--the marginal utility with respect to X depends on X again--is going to be one-third times one over X because the derivative of the log is one over X, and as X gets bigger that also declines. So these are the two functions that we're going to use over and over again because I want to make things concrete with actual numbers. So we'll always solve examples with quadratic stuff, maybe everything will be quadratic or linear, and with logarithmic stuff.
Those are the only two functions you really have to be totally comfortable with. So you have to understand what a derivative is. This is a partial derivative. So how much extra utility do you get out of consuming more X? If you've already got a certain amount of X in your possession it's a hundred minus X. How much more utility do you get out of consuming more X? If this is your utility when you're consumption's already a certain amount of X it's one-third times one over X. So those are the two things you have to be comfortable with using. So that's utility.
What else do we need to describe a person? It's his endowment. So with only two goods, so here's X and here's Y, so we could have an endowment EiX, EiY. That's the endowment of X and Y of a certain person, EiX and EiY. So this person, let's say it's this top guy--a hundred X minus one over two X squared plus Y--he has a certain utility function, he's got a certain endowment. Maybe there's somebody else over here who I can put in a different color. Aha, I think pink is a good color. So another person might be over here and this is EjY and EiX. So J has a lot more of Y, and I has a lot more of X. They're two different people, but you could imagine not two people you could imagine 150 of you with different endowments and different utility functions, or 300 million of you with different endowments and different utility functions.
And what general equilibrium is about is saying, well, if you've got all these people with well defined utility functions, those are the data, we may not know them but they know them themselves with all those utility functions and all those endowments, and you throw 300 million of them together, or 150 of you together, can you predict what's going to happen and is the thing that happens good for the society. So that's the problem of general equilibrium.
And it turns out that with these simple utility functions it's very easy to solve for equilibrium, predict what'll happen, and things look great until you get to financial equilibrium. And we'll be able to solve them either by hand or on a computer, and we're going to take advantage of that because we want concrete answers to concrete problems, and we want to interact it with the financial world to see what happens. So remember, what's the next step? The first step is exogenous variables. So we define the exogenous variables. The next step is endogenous variables. So what are the endogenous variables going to be? And the endogenous variables are going to be the prices and the trades, or final consumptions.
You can always deduce a trade from a final consumption because if you know your endowment, the exogenous thing, and you're consuming more of X than you're endowed with you must have bought that difference somewhere. And if you're consuming less Y than you started with you must have sold some of that Y in order to end up consuming less. So the endogenous variables are the prices and the trades.
Now, how can we make a general theory that for an arbitrary number of people, an arbitrary number of goods, you can solve and figure out what's going to happen that looks very much like the example and has as a special case the example we did to begin with? That's what happened with general equilibrium, and I'm about to describe it. So the next step is always to write down the equilibrium as a bunch of simultaneous equations. So what are all the equilibrium equations going to be, and that's what's going to be our model of what happens in the world.
Are there any questions? How are you all doing here? Is this painfully repetitive of what you know. I need some feedback here. How many of you haven't seen this before? Everybody's seen this before? What about all these people who e-mailed me and said they were scientists and philosophers and psychologists and they wanted to take economics the first day. So you're one of those people. Maybe you didn't e-mail me. So this is a first for you, but everybody else you've all seen this before.
Well, that's good. I can move along here. So I'll keep looking at you as I proceed here. So don't feel bashful. Speak up if it's not making sense. So what was the great conceptual advance? It was--one conceptual advance was the budget set. Now, this will turn out to be, in economics--the rest of the 140 of them have all got this down, but as soon as we turn it into a financial problem they're not going to be able to do it again even though it's going to be the same idea. So this budget set was an extremely clever idea which I'll now repeat for them and tell you for the first time, but I can almost guarantee that although they all think it's obvious, when we do the first financial problem they aren't going to be able to do it even though it's the exact same idea. So what's the idea?
You begin with your endowment, EiX and Eiy. So this person has to buy and sell X and Y. So the person says to himself, "I've started with this X and Y, I might like something that's better." Now how can you illustrate what's better for this person? Well, Edgeworth, as I mentioned, Edgeworth invented the idea of the indifference curve. So he says, "All the goods that are of the same utility can be described by this indifference curve X." This person, her utility is one-third log X plus two-thirds log Y, well if she consumes less of X, enough extra Y will make her just indifferent to where she was before because there's a tradeoff between X and Y. Economics is all about tradeoffs. So this is her indifference curve. Maybe his indifference curve looks like that, a different slope, entirely different.
So he thinks a lot of Y. A little diminution in Y you better get a lot of X to compensate him. She's kind of more balanced in things, X and Y, unless she starts to get too much of X in which case Y is more important to her. She in general is more balanced than he is. But anyway, so they have different tastes, different utility functions, and different endowments. So what's going to happen in the end?
Well, the budget set describes what she can do. We're going to assume, as we did before, that cornerstone of economic reasoning, somehow when these hundred million people, 300 million people get together they're going to miraculously discover the price. They're going to be screaming at each other, but we don't care about that. We just say for the purpose of the big picture, some price of X and Y is going to emerge. So equilibrium is going to be a price of X and a price of Y. It's going to emerge and now what can she do? Well, she can say, "Given my X I can buy more X than I started with, and if I do that the price of X is PX." So if I want to buy more--I have this already. So I want to end up consuming Xi, so final consumptions will be Xi and Yi, this is the final consumption, so my trade, if I want to buy more, I can express the idea that I'm buying more by saying my final consumption is bigger than my endowment. So I've had to buy, I've had to trade to get this much more which means I had to pay PX times this difference.
Now, how did I get the money for that? Well, I got the money for that by selling some of Y. So I sold Y. I started with EiYand I sold some of it because I ended up with less than I started. So the money I got by selling Y I can use to spend on buying X. That's the basic budget constraint. Now, the cleverness is in realizing that it doesn't matter which one--So here Xi is bigger than EiX. You're buying X. Here Yi is less than EiY. You're selling Y. And so the revenue you get from selling Y equals the expenditure you make on buying X.
So the cleverness is in realizing it doesn't matter what the signs are. If Xi is less than EiX this equation still makes sense because then you get a negative number. You've gotten money by consuming less X than you started with so that's money you can use to buy Y. And then Yi--you'll be able to buy more Y than you started with so this number will also be negative by the same amount as this. This is the extra value on Y. This is the extra value on X. So whether the X's and the Y's are bigger or smaller than the EiX's or EiY's this equation defines the budget trading opportunities of the agent. Did that go too fast? You got that.
So you can write that a little bit more simply by saying, putting a plus here and reversing the order, making it more symmetric. So this is Yi minus EiY equals zero. So that's the budget set of agent i. And in the diagram the budget set--I'm out of colors that show up I think--all the others got vetoed, I think orange was okay--the budget set, then, will be something that looks like this. That looks terrible. How bad can you get? So that budget set might look something like--let's make it this way. It looks something like that. It's a linear line that goes up--just forget this guy's budget set. We'll do the other one. I can get it better in the picture. So this one’s budget set, his budget set might look something like that. So his budget set, never mind hers, it goes off the page, his budget set he starts with this endowment.
If the prices are given PX and PY, PX and PY define a linear tradeoff between Xi and Yi, in this case j, because the more X you consume the less Y you have to consume and there's going to be a linear tradeoff between the two given by rearranging these terms. PX and PY are fixed, so this is just a linear equation in Xi and Yi, and so that tradeoff is given by that budget set. So mister pink is going to try, given his opportunities on this budget set, to pick the combination of X and Y that's best for him. And so that's going to turn out to be something that's right here because no other combination of X and Y will give him as much utility as that.
Did that make sense? All right, so that's it. That's the main lesson. So how do you describe now the whole equilibrium conditions? Well, so equilibrium now, if you can see this, equilibrium is defined by what? It's defined by PX, PY, and Xiand Yi for all i in I. It's just the prices that emerge and final consumptions that everybody chooses of X and Y. There are only two goods here. So the price of X, the price of Y what every person i ends up with Xi and Yi, and what has to be the case? What has to be the case?
The first equation is going to be that the final consumptions of everybody have to equal the final endowments because everyone who buys has to be met by another seller. Remember equilibrium was price taking, agent optimization, rational expectations and market clearing. Price taking means everybody knows what the prices are, miraculously PXand PY, before they act. Agent optimization we're going to come to. It means they do the best thing they can. Rational expectations means even though they're only buying one good and there are thousands in the economy they understand all the prices, and when they act they're taking into account all of the tradeoffs they could make. So they realize the whole vector of prices. And market clearing means for any buyer there's a seller, so market clearing means summation from i in I of Xi has to equal summation i in I of the endowment, EiX of X.
So in this picture if I added this to this, this is the endowment, so I add this vector to that vector I get this thing over here, and this is going to be the total endowment in the economy. So this total endowment EiX, I add over every person i what the total endowment is. So I add his endowment of X to this guy's endowment of X and I get the total endowment of X. I add her endowment of Y to his endowment of Y and I get the total endowment of Y. So the first two equations are summation i in I. Yi equals summation i in I of EiY. The third equation is everybody is going to choose on their budget set.
Everyone, this person--mister pink here--he's going to choose not inside his budget set, he can't choose outside of it because there's no point in wasting money. He's going to buy the combination of X and Y that lies on his budget set that does as well as he possibly can. So the equation here is going to be that PX times Xi minus EiX plus PY times Yi minus EiY is equal to zero. I could do this for j too just since I've got a picture of--this is PX, this is PY. PX times Xj minus EjX, so I'm doing a special case now with two people, Yj minus EjX equals zero. Everybody's on their budget set. So he's on his budget set, she's going to be on her budget set.
Her budget set, by the way, is better than his because her budget set is going to look like this, right? It's got to be parallel to his because the prices she faces are the same and her endowment is worth more than his. So her budget set is further out. So that's what he does, that's what she does, or that’s what she does, that's what he does. And now the fifth one--so now we have the two mysterious equations that are left.
So how do we express the idea that the choices Xi and Yi by i, that's her--and she's going to optimize by choosing here somewhere. This is her indifference curve, right, looked like that. So that's what she's going to do. And remember he's going to choose here. So how can you turn her choice and his choice into an equation? Well, this was invented by a German guy Gossen in 1851 and then rediscovered by Jevons, Menger and Walras, the same three I mentioned several times now. This is the marginal revolution in economics. What they said is you can turn the behavior of individuals, of humans as Gossen said, "I can do for the bodies on earth what Copernicus did for the bodies in heaven, find equations that describe their motion."
What is it that people are going to do? To say that you're choosing the best possible thing means that the slope of the budget set is equal to the slope of your indifference curve, but what is the slope of your indifference curve? That's the tradeoff between X and Y. So what does it mean? If you get a little bit less X you're losing the marginal utility of X. If you get a little bit more of Y you're gaining the marginal utility of Y. If the price of X and Y are the same then it had better be that the marginal utility of X is equal to the marginal utility of Y because you can always give up one unit of X and get one unit of Y.
If this is optimal, and you can give up one unit of X and get two units--sorry, if the marginal utility of Y was double the marginal utility of X then you would give up that one unit of X and you'd get two extra utils by taking the one unit of Y which you can afford by selling one unit of X, and the utility would be much higher than it was here. And so you wouldn't be optimizing by doing that. So the final equation is you're optimizing if and only if the marginal utility of i of X divided by the marginal utility of i of Y equals PX over PY. And the last equation is the same thing for j, the marginal utility of j of X divided by the marginal utility of Y has to equal PX over PY. So why is that again? That's the trickiest equation.
That's the one that Marx and Adam Smith and not even Ricardo, the most brilliant one of them all, not even Ricardo could figure that out, this equation marginal utility, wait until 1871. And again, to repeat it, it's of course very obvious now but wasn't at the time, how can you describe what these people are doing? You have to figure out the budget constraint, that's what they can afford, and then they're going to choose the point on their budget constraint which maximizes their utility.
But that just means in the picture it makes the indifference curve tangent to the budget set, which means that you set--so and what is the slope of the indifference curve? Well, the tradeoff between X and Y that leaves you indifferent--how much X do you have to give up to get an extra unit of Y and still be indifferent? It's determined by the ratio of the marginal utility of X to the marginal utility of Y because those are the, you know, when you give up a unit of X you're losing the marginal utility of X. When you're getting a unit of Y you're getting the marginal utility of Y. If you can trade them off in the market at 3:1 you optimize when, in your own personal evaluation, you're trading them off on the margin at 3:1. You really follow that?
That's an idea that took fifty years to figure out and you claim you figured it out now in five minutes, so that's good. So you'll have a chance in the problem set to get practice. So those are the equations. We now basically have described economic equilibrium. So we now have the ability to play with all kinds of models, as we'll start in the next class doing, solving for economic equilibrium, figuring out what will happen, and then complicating it by adding a financial sector and see how that affects what goes on in equilibrium.
[end of transcript]