ECON 251: Financial Theory

Lecture 5

 - Present Value Prices and the Real Rate of Interest


Philosophers and theologians have railed against interest for thousands of years. But that is because they didn’t understand what causes interest. Irving Fisher built a model of financial equilibrium on top of general equilibrium (GE) by introducing time and assets into the GE model. He saw that trade between apples today and apples next year is completely analogous to trade between apples and oranges today. Similarly he saw that in a world without uncertainty, assets like stocks and bonds are significant only for the dividends they pay in the future, just like an endowment of multiple goods. With these insights Fisher was able to show that he could solve his model of financial equilibrium for interest rates, present value prices, asset prices, and allocations with precisely the same techniques we used to solve for general equilibrium. He concluded that the real rate of interest is a relative price, and just like any other relative price, is determined by market participants’ preferences and endowments, an insight that runs counter to the intuitions held by philosophers throughout much of human history. His theory did not explain the nominal rate of interest or inflation, but only their ratio.

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Financial Theory

ECON 251 - Lecture 5 - Present Value Prices and the Real Rate of Interest

Chapter 1. Implications of General Equilibrium [00:00:00]

Professor John Geanakoplos: I think I’m going to start. So this is really the beginning of the finance part of the course. So far we’ve reviewed general equilibrium, which I said Fisher invented or reinvented in order to do finance. And as you remember the main conclusions from general equilibrium are first that the market functioning by itself without interference from the outside, in other words a situation of laissez-faire, leads to allocations that are Pareto efficient.

So they’re in some sense good for the economy and good for the society. They don’t maximize total welfare. That’s not even a well-defined thing as we saw last time because how can you measure, how can you add one person’s utility to another. It doesn’t even make sense. So economists at first were wrong to think of that as the criterion for good allocations, but there’s another better definition of efficiency that Pareto invented, called Pareto Efficiency, and the free market achieves Pareto Efficiency at least if there are no externalities and there’s no monopoly.

So that, lesson number one, was taken to mean that the government shouldn’t interfere in the free market, especially shouldn’t interfere in financial markets, and that’s something we’re going to come to examine.

The second lesson we found was that the price is determined by marginal utility. It’s not determined by total utility. So it may be that water is much more valuable than diamonds because it does a lot more good for everybody and for the world as a whole than diamonds do, but the last drop of water, really most people have as much water as they need, the last drop of water is not doing that much whereas the last diamond is a rare thing and not many people have them. So the last drop of water is worth less than the last equal weight of diamonds and therefore water is much cheaper than diamonds even though water’s much more valuable as a whole than diamonds. The price of things depends on their marginal utility.

A third implication of what we did is that there’s no such thing as a just price. It depends on what peoples’ utilities are and how much they like it. It depends on how much of the good there is. That’s why diamonds are priced less than water [correction: that’s why water is priced less than diamonds]. And it depends on how wealthy people are. If you transfer money from people who don’t like apples compared to tomatoes, to people who like apples a lot compared to tomatoes, the price in the free market is going to reflect more the latter class of people than the former because they’ve got the money to spend, and so the price of apples is going to go up relative to the price of tomatoes. So those are the three basic lessons of general equilibrium.

The first one about laissez-faire has a huge implication for whether there should be regulation, but the second pair of implications, what determines the price and how price changes as you redistribute wealth and so on, and no just price, that set of ideas, you’ll see, is also going to be very important for finance.

Chapter 2. Interest Rates and Stock Prices [00:03:08]

So those lessons seem clear. Some of those lessons were understood already by Aristotle as we said. So the ancients understood supply and demand, at least a little bit of supply and demand, and yet as soon as they moved from apples and oranges to finance, they all got hopelessly confused. So Aristotle said, “Interest is unnatural.” I could go through a lot of people and what they said, but I’m going to just leave it at a few quotations. The Bible says interest is terrible. Judaism frowns on interest. Christianity frowns on interest. Islam frowns on interest. All the great religions of the world crystallizing, obviously, some of the most important thinking of the time, frowns on interest, so just to remind you of a few. So why do they frown on interest?

Well, the idea is that you do nothing, the lender does nothing and he gets back more than he lent to begin with. He’s making a profit without having exerted any effort whatsoever. So, Sulinay Middleton said, “In trade both parties are expected to gain, whereas in lending at usury only the usurer could profit.” So in Deuteronomy in the Bible, so this is the Jewish Bible, is says, “Thou shalt not lend on usury,” (that just means interest). “Thou shalt not lend on usury to thy brother. Usury of money, usury of victuals, usury of anything that is lent upon usury, “–that’s all terrible. Of course, “unto a foreigner thou mayest lend upon usury, but unto a brother thou shalt not lend upon usury.” So the Jews could lend to Christians but not to each other.

So the Christian Church outlawed usury, called it a mortal sin. Luther, for example, says, “For who so lends that he wants it back better or more, that is open and damnable ocker. Those who do that are all daylight robbers, thieves and ockerers. Those are little Jewish arts and tricks.” So there was this antipathy towards usury, and because Jewish moneylenders were able to lend to Christians there was an antipathy to Jewish moneylenders which we’re going to come to when we talk about Shakespeare. So Muslims also forbid lending.

In fact, even today it’s illegal to charge interest in Islamic law. So in my hedge fund we tried to raise money, and there’s lots of money in the Middle East, and most of it, by the way, ten years ago, almost all Middle Eastern money was invested in U.S. Government bonds and U.S. stocks, nothing else like in mortgages, for instance. So I went to Saudi Arabia and I met a bunch of brothers of the King, the eldest brothers of the King, and I suggested they invest in our hedge fund.

And they actually became sort of interested, and so we had to write up a complicated contract. Now, you know a mortgage pays interest, so if you invest in the hedge fund and the mortgage is paying interest it looks like they’re getting interest, and so that wasn’t going to do. So we had to write a very elaborate contract which disguised the fact that interest was being paid, and it had to be overseen and blessed according to Sharia Law by a holy person who was going to verify that there was no interest. Now he charged a fee which was a percent a year which looked an awful lot like interest, but anyway so.

So the point is all these religions have banished interest despite the fact that they themselves were involved in interest, and lending, and borrowing. A world can’t function really without lending and borrowing and the charging of interest. So these religions that forbade it at the same time knew that it was going on and allowed it to go on and sometimes participated in it.

But the point I’m trying to make is that there was vast confusion, and even today there’s confusion because still today the Jewish law doesn’t allow for interest between Jews, and–there’s a charade that goes on there just like there is in Islamic law, and just like there is–still frowned upon by the Christian Church. So it’s a hard subject to understand, and why is that? Why is it that it’s so confusing, and how should we understand it? Well, Fisher cut through all this extremely simply, and the way he did it was he said, “Let’s just think mathematically then we won’t get so tied up in all these religious complexities. Just let’s do something mathematical and concrete.”

So suppose that we consider a problem, which is the one I’m going to work with the rest of the class. Maybe I better do it over here. So let’s say that there are two agents and two goods. So the two goods now are X1 and X2. So Fisher’s first insight is that let’s think of X1 and X2 as apples, but apples today and apples next year, Fisher said–although they’re both apples, exactly alike, there’s no difference between these apples, the apples this year are different goods from the apples next year. So let’s move away from apples and tomatoes to apples this year and apples next year. So I’m not going to call them goods X and Y anymore. I’m going to subscript them by time.

These are both X because it’s the same good, but they’re different goods because they occur at different time periods. So Fisher said we can incorporate time simply by having different goods. So of course people, he said, are going to have some utility of consuming today versus consuming tomorrow, and let’s say this utility is (log X1) + (1 half log X2) for Mr. A. So I’m going to come back to this in half an hour and explain why Fisher thought that this half made sense.

You see, this Agent A likes good 1 a lot more than good 2. So Fisher would say that’s because Agent A is inpatient. An apple is an apple, but if you get it now it’s worth more to you, it gives you higher utility than getting an apple next year. “This is a law of human nature,” he claimed, which I’m going to come back to later, and that’s why when you write down the utility function there’s a discount factor, which we’re going to add–a discount factor–which discounts, reduces, the utility you get from future consumption. So let’s say (UB of X1 and X2) = (log X1 + log X2). So B is more patient than A is. B actually doesn’t discount the future. A does discount the future. So A is impatient, relatively impatient, and B is patient.

They have endowments, so EA let’s say the endowment is (1, 0). Say it is (1, 1) and EB let’s make that (1, 0). But now Fisher wants to talk about finance and he wants to talk about stocks, and bonds, and interest and all kinds of things. So he says, “We’ve talked about good with no problem. We can talk about goods today and next year with no problem, let’s talk about stocks.”

What is a stock? Let’s say there are two stocks, stock alpha and stock beta. What are stocks? I mean, they’re pieces of paper that you’re trading, but they give you ownership of something like a factory or a company or something, and what good is the company? Well the good of the company is that it’s going to produce something. So let’s say that the stock is going to produce something in the future. So we’ll call the production of the stock–so what is the future?

There are only apples in the future. So let’s say Dalpha2 is 1 and Dbeta2 = 2. Fisher says, you can tell a lot of stories about what this stock does, and what its method of production is, and what kind of managers it has and a lot of stuff like that, but in the end people care about the stock because the stock is going to produce something, and the value of the stock is going to come from what it produces. So Dalpha2 is what people expect the output of the stock to be next year which is the last year we’re worrying about, and Dbeta2 which is 2 is what people expect the stock to produce next year.

And we’re going to assume that perfect foresight here. So Fisher says, “Well, in general people’s expectations might be wrong,” but let’s start off with the case–people anticipate something, surely they’re looking ahead to the future when deciding whether to buy the stock. We’ve got to assume something about what they think. Let’s suppose they actually get it right and they know what the price of the stock is next period. So what’s going to happen?

Well, we can define an economy and presumably the interest rate and the stock prices and all that are going to come out. Now I should mention, by the way, I forgot to say this, but as I write this down I suddenly realize I forgot to mention it. There are other theories of interest too. Another famous one was Marx’s Theory of Interest. So this is to be contrasted with Fisher. What did Marx say? So in my youth when I was your age it was very fashionable to be a Marxist. You had to study Marxism basically. If you wanted to talk to women you had to know about Marx. So anyhow, I dutifully went off and read Marx.

And so what’s the idea of Marx? The idea was that he imagined an agricultural economy where you plant stuff today and then the output comes out tomorrow. So you put corn in today, corn comes out tomorrow. So it doesn’t require much effort to plant the corn. You have to buy the corn. So the capitalists would buy the corn, but planting it didn’t require much effort. However, harvesting it, picking the cotton, picking the chocolate, picking all that stuff takes a lot of effort. So in the end you’d get a lot of output.

Now when you pick the output you’d have to pay workers in order to pick the output. So Marx imagined that there was a wage that was arrived at by the struggle, class struggle, between the capitalists and the workers over the subsistence wage. So the subsistence wage was something that resulted from this huge class struggle, and over time maybe it would rise as workers got stronger, but it was still always quite low and the subsistence wage was what the workers would get. And what’s left over, which was the surplus–so the output being more than what was put in, was the surplus. Part of the surplus would go to the subsistence wage. The rest would go to profit.

And so if you look at how much was put in to begin with, you get all the output back out, the same amount of corn you put in plus some extra you have to give to the workers, and extra that the capitalist gets back as his profit. The fact that the capitalist has done no work at all, he’s just bought the corn, let someone else plant it, let someone else harvest, paid all those guys virtually nothing at the beginning and a lot at the end, he’s gotten profit for doing nothing just like when you lend money, so the profit divided by the initial outlay, that was the rate of interest. And so Marx said that a capitalist, he could put his money in his bank or he could run this farm and make profit this way.

So the money, interest in the bank would have to turn out to be the same as this rate of profit otherwise he’d put all the money in the bank. And if it was smaller the banks would have to give higher interest in order to attract depositors. So the capitalist’s profit–rate of interest was determined by the rate of profit and the rate of profit was determined by the struggle between capital and labor. So we’ve got these religious figures and great philosophers saying interest is terrible. We’ve got this great philosopher-economist saying it’s the result of a class struggle, and now we’ve got Fisher, actually Marx was pretty mathematical, but now we’ve got Fisher turning it into a simple math problem and saying, “Let’s reason out the math problem and we’ll have the answer to these questions, and it’ll turn out to be quite different from what all these guys are saying.”

So here’s his economy that I just described. The Fisher example, not literally an example he gave, but similar to one he gave. So he said, “All right, what happens in this economy? Let’s just be very commonsensical. What we need to find out now is financial equilibrium,” so financial equilibrium is much more complicated seeming than we had before, because we care about the prices. So now I’m going to use (q) for prices, a q for contemporary prices, qcontemporary, so the price you pay today to get the apple today. q2 is the price you pay next year to get the apple next year. They’re contemporary prices.

And of course people are going to decide what they want to do, XA1, what they’re going to end up consuming XA2, XB1, XB2, but now we’ve got a more complicated world. There’s stocks to be traded and there’s the price of stocks. So P–pialpha, I’m running out of letters. I’m going to switch to a Greek one. This is the price of stock A, stock alpha. And pibeta is the price of stock beta. And then we have to know how many shares are they going to hold? Well, it’s going be thetaA, A’s going to hold a certain number of shares of alpha, and thetaAbeta, A’s going to hold a certain number of shares of stock beta, and B’s going to hold a certain number of shares of alpha and a certain number of shares of beta.

So we want to solve for all of that. Now, I should have said at the beginning if these are trees producing apples there was an initial stock. People owned a certain number of trees. So let’s say theta-baralphaA, so this is the original ownership of alpha. Let’s just say that’s, I’ll make up some number. I might as well use the same number I thought of before. Let’s say that’s 1, and let’s say theta-barBalpha is 0, and let’s say theta-barA of beta is a half, and theta-barB of beta is a half. So the original economy is more complicated than before because we’ve added stocks.

We characterized the stock. We said a stock is a very complicated thing. A company is very complicated, depends on managers and processes and there’s all kinds of stuff you think about when you think about a stock, but really at heart all people are trying to do is forecast what are they going to produce. And so we’re going to make it simple mathematically and say let’s say we know what they’re going to produce next period. So let’s say it’s a tree.

Everybody knows the alpha tree’s producing 1 apple. The beta tree’s producing two apples. Alpha happens to own the only alpha tree. A owns the alpha tree, excuse me, and A and B own half each of the beta tree. So that’s the original economy and the equilibrium is going to be, what are the prices going to turn out to be of X1 and X2, what are the prices of the trees going to turn out to be, how much will people consume and how many shares? Because alpha A began with all of tree maybe he’ll sell his shares of tree and end up with not having a tree in the end. So we have to see where they began, the stock ownership to begin with and where they end up. So we have to solve all of that and it looks way more complicated than before, and so complicated that you can see why people might have gotten confused.

But according to Fisher it’s going to turn out to be a very simple problem in the end once we look at it the right way. So are there any questions about what the economy is and what are the variables that we’re trying to explain? Yeah?

Student: Sorry, I can’t read what that says over theta alpha A. That’s original ownership?

Professor John Geanakoplos: Original ownership of stock alpha. And this is original ownership by B of stock alpha. This original ownership by A of stock beta, and this is the original ownership by B of stock beta, original ownership of alpha. Thanks. Yes?

Student: Having defined all of these could you redefine what D stands for?

Professor John Geanakoplos: D is the dividend. That’s the output that we can all–thank you. I should just write this down. This is the anticipated dividend, which is the output since that’s the end of the world of stock alpha in period 2. And Dbeta of 2 is the anticipated dividend of stock beta in period 2. So it’s 1 apple we’re getting out of the alpha tree, 2 apples out of the beta tree, right?

That’s what the tree’s good for. We can look at how beautiful it is. We could talk about how much the owner’s actually watering the tree. We can talk about a lot of complicated stuff, but in the end all we care about is how many apples we expect to get out of it. All the other stuff goes into helping us think about how many apples we’re going to get out of it in the end. So we cut to the bottom line. What are the apples we expect to get out of tree, 1 from the alpha tree, 2 from the beta tree. Someone else had their hand up.

Student: I had it, but you answered it.

Professor John Geanakoplos: Any other questions about this set up?

Chapter 3. Defining Financial Equilibrium [00:22:06]

So we’re returning to first principles here, very simple example. When there’s ever a big confusion about something important it’s always good to go to first principles. There was a chess player when I was young named Mikhail Tal who was a world champion for a little while, and he said that every two or three years he’d go back and read his original introductory textbooks on chess. So we’re going back to the first principles.

How would you define equilibrium here for a financial equilibrium? Well, the first thing is just common sense. What are people doing? At time 1 what can they do? They can spend money, so I’m going to look at the budget set for Agent i, and i can be A or B so I don’t have to write it twice. So he’s going to say to himself, let’s say A is he and B is she, i is he. He’s going to say to himself–let’s say i will say, “How much does it cost me to buy goods?” Well, the cost of apples is q1 times X1. That’s how many apples I might end up with. Now, how much does it cost me to buy shares? It’s going to be pialphatimes how many shares I end up with, thetaalpha, plus (pibeta times thetabeta).

So I’m buying goods, I’m buying alpha shares and I’m buying beta shares and this is how much I have to spend to get the holdings I want of each. Now where did I get the money to do that? I got the money to do that because I started with my endowment of goods which was Ei1, which in this case for A was 1 unit, for B was also 1 unit, and then I also had shares to begin with of these stocks. So I had ([pialpha times] theta-baralpha) + (pibeta theta-barbeta). In period 1 that’s what I had to do. I wanted to buy apples, shares and I had shares to sell and apples to sell.

So that’s what i did. So of course if X1 is bigger than the number of apples i started with that means i has bought apples because he ends up with more than he started with, so that meant he must have been buying apples. If thetaalpha is more than theta-baralpha it means that alpha [correction: i] bought shares of stock alpha. Thetaalpha is less than theta-baralpha it means alpha [correction: i] sold shares of stock alpha.

All right, now in the second period what happens? Well, in the second period we have q2 times X2. The shares are going to be worthless in period 2. So no one’s going to buy them. Why are the shares worthless? Remember that when you buy a share of stock the dividend comes later. You don’t get the dividend immediately. So someone buying stock in period 2, it’s too late to get the dividend. It’s already gone to the owner who bought the shares in period 1. So the buyer of a stock [doesn’t] gets the dividend for a month or something. So next period’s dividend is still going to go to the buyer in period 1, that’s why it’s valuable to buy shares in period 1 because you get next year’s apple.

So by next period you can buy the tree, but the world’s coming to end. That tree’s not going to do you any good. It doesn’t produce any more apples. So nobody’s going to bother buying shares. I don’t have to bother with them. The prices are zero. And so what income do people have in period 2? Well, they’ve got the contemporaneous price times the apples that somehow they find on the ground or that their parents are going to leave them when they get old. So that’s their endowment of apples, but what else do they have? They’ve got more apples than that. What else do they have?

Student: The dividends.

Professor John Geanakoplos: The dividends. So what are the dividends? Well, you bought thetaalpha to begin with so that’s Dalpha2. So if you bought the whole tree then you’ve got all the dividends, and similarly with beta. Thetabeta times Dbeta2 [correction: thetabeta times Dbetatimes q2. And, thetaalpha times Dalpha2 should also be multiplied by q2]. So that’s it. So the budget set is a little more complicated. So that’s the budget set. So it’s got 2 equalities instead of 1 equality, so already things look a little more complicated. Now, so an equilibrium is going to have to be that i chooses (Xi1, Xi2), thetaalpha–I can write, thetaialpha, thetaibeta, that’s all the choices he has to maximize Ui subject to this budget set.

So A’s going to pick what shares to hold, how much to consume today, then, of course, looking forward A’s going to be able to figure out what he’s going to end up consuming tomorrow. All right so, and now in equilibrium we have to have that (XA1 + XB1) has to = (EA1 + EB1). And then we do the shares, thetaAalpha + thetaBalpha has to = theta-barAalpha + theta-barBalpha, right? The stock market has to clear and thetaAbeta + thetaBbeta has to = theta-barAbeta + theta-barBbeta. So in period 1 the demand for apples has to equal the supply of all the agents, but now what’s the last equation? This is a little trickier. What’s the last equation? XA2 + XB2 = EA2 + EB2 +…. Is that it? No. There’s something else.

Student: Plus dividends.

Professor John Geanakoplos: The total consumption of apples is going to be the apples that they have on the ground, but also the ones that were picked off the trees, so these dividends. So it’s going to be the total dividends which are [(theta-barAalpha + theta-barBalpha) times that tree–Dalpha2] + [(theta-barAbeta + theta-barBbeta) times Dbeta2]. So just to say it in words, it’s exactly what we had before except we have to take into account in addition to the goods market clearing, we have to take into account that the stock market has to clear.

And in the end demand for goods has to equal the supply that people had in their endowments, but also what the companies are producing. These companies are producing output, apples. And so that’s part of what the consumption is going to be in the economy. Are you with me here? It’s a good time for questions, maybe. Yes?

Student: Could you just explain again why we don’t take stocks into consideration in period 2?

Professor John Geanakoplos: So in period 2 you might wonder, pialpha is the price of the stock at period 1, stock alpha in period 1. Pibeta’s the price of the stock beta in period 1. How come I didn’t write down the price of the stocks in period 2 and keep track of what they’re holding in period 2? And the reason is that when you buy a stock you’re buying it not for the dividends at this same moment in time. You don’t get those dividends. The guy who already had it gets those dividends.

When you buy the stock you’re buying it for the future dividends you can get, and I’ve assumed the world’s going to end after 2 periods because nobody’s utility cares about period 3. So if you buy the stock in period 2 it’s too late for you to get anything. There are no dividends because you can only get them in period 3, and there won’t be any dividends in period 3, and if there were you wouldn’t care about them anyway. So the stock’s worthless to you. So the price of the stock in period 2, of both stocks, is going to be 0. So there’s no point in putting down what people are buying of the stocks or selling or anything. It’s just not worth anything.

But in general you’re right, and we’re going to be more complicated later when you look at your income from having bought the stocks. You’ll have as your income the dividend from the stocks plus the resale value of the stock, because you could sell the stock next period. But I just know the resale value’s going to be zero because you’re in the last period of the economy. And I just want to keep it very simple this first time. Step by step you’ll be able to keep very complicated things in your head but not right at first, so any other questions? Yes?

Student: Why are there endowments in period 1 and <>?

Professor John Geanakoplos: You mean why do people have endowments today and next year? Because you could think of the endowment, for example, as–here we’ve got apples, but usually the endowment is your labor, so you can work this year–your most important endowment is your energy and your labor. So you’ve got it this year. Next year if you’re still alive that’s new labor that you have. It’s a different good so it’s a second endowment that you have. So I don’t want to get caught up in labor and all that and get involved with Marx again, so I’m just going to talk about apples.

You have an endowment of apples when you’re young and next year somehow you’re going to have more apples. So you might have thought that the only apples next year come from what the firms are producing, but I allowed for the possibility that people have apples too just like their labor next year. Other questions? Yeah?

Student: We then have to define E2A and E2B in terms of first period endowments, or is that something implied in the equation?

Professor John Geanakoplos: I should have written this more carefully maybe. EA1 and EA2 is that, and this is EB1 and EB2 is that. So as he was suggesting back there I’ve assumed for person B that he’s got an apple now. We aren’t modeling what happened to get us here. The guys got an apple today. They both have an apple today. Somehow A’s also going to have another apple tomorrow that he’s going to find under his doorstep somehow that isn’t being produced by the tree.

And maybe you can think of it as labor that he’s going to have next period.

Chapter 4. Inflation and Arbitrage [00:33:41]

All right, so that’s it. Fisher says as soon as you write down the economy mathematically all sorts of things are going to occur to you which if you’re talking in words about justice and injustice you’re going to be lost. So what can we get right away out of this? What can we get right away out of this?

Well, the first thing is, how would we define inflation? What is inflation? What’s inflation in this economy? Assuming we’ve got the equilibrium, which we’re going to get soon, we’re going to calculate it, but right now we don’t what the numbers–you know we’ve got a bunch of equations and stuff. We don’t know what XA1 and q1 and q2 are going to turn out to be, but we’re going to find out very soon. But before we find out, assuming we’ve gotten those, what will inflation be? What is inflation? Yeah?

Student: Is it how much the ratio of the price of the dividend has changed?

Professor John Geanakoplos: Well, we’re talking about inflation. When you talk about the Consumer Price Index, inflation, what are they talking about? Yeah?

Student: It’s the rise of q1 and q2.

Professor John Geanakoplos: So inflation is just q2 over q1, right? So that’s the price of apples today. That’s the price of apples next year. If the price of apples next year is bigger than this year we’ve got inflation. If it’s lower we’ve got deflation. So already the model, you’re talking about inflation. What else? What’s the next most obvious?

Well, I think I’m going to skip a bunch of stuff and get now to the key idea. The key idea is arbitrage. So Fisher says, “People have foresight.” They’re anticipating what the dividends are going to be. They understand that you can talk about how beautiful the tree is, and how much you like the owners, and how much they’re watering it, and whether they have a good plan for irrigation, and whether they did well in college and stuff like that, but in the end all you care about the trees is how many apples they’re going to produce. So knowing that, can we say something about pialpha pi alpha versus pibeta? In equilibrium what’s going to have to happen? There is going to be some connection, and what’s the connection going to be? You’ve got two trees. Yeah?

Student: The ratio between the prices would be the ratio between the <>?

Professor John Geanakoplos: Right, so pialpha is going to be pibeta times D[alpha2 over Dbeta2]–alpha will be better as long as the–hopefully I’ve got that in the right order. And so in this case pialpha is going to equal a half pibeta, because alpha’s producing half the dividend that beta’s producing. So obviously it’s going to turn out to have half the price. That’s the fundamental principle.

We’re doing it in the most trivial case, but it’s the most fundamental principle of finance that if you’ve got two assets and they’re basically the same up to scale then their prices have to be the same up to scale. Who’s going to bother to buy alpha if it costs the same amount as beta when it only produces half as much? Yes?

Student: Is that the same thing as saying that their yields will converge, that equation?

Professor John Geanakoplos: Well, it is something like that, but that’s a word that we haven’t defined yet, so we’re going to define it in the next class. So it’s something like that, yep. Any other? So that is a very simple thing. Suppose after finding the equilibrium I added a third asset that paid 1 dollar in period 2 next year. Now, it would have a price of–added a third asset gamma, so pigamma–we’d have to solve for the equilibrium pigamma, and is there some word that I could use? So gamma is an asset that pays a dollar in period 2. It’s like a bond promising a dollar in period two. The price of the bond would then have to be what? 1 over (1 + i) where i is called the nominal interest rate, so we’ve got inflation is occurring in the model. If I added a bond, which I didn’t bother to do because it’s just yet another thing I’d have to write down, I could have had a third asset which pays a dollar.

The others are paying off in apples. This one’s paying a dollar and its price today, if you pay 80 cents today it’s like saying I’m paying 80 cents today, I’m getting a whole dollar tomorrow. So it’s like a 25 percent rate of interest, because another way of saying it is that 1 over pigamma is 1 + i. I put only pigamma in today, I get 1 out tomorrow so I’ve gotten back not only the pigamma I put in but something extra, that’s 1 + the interest rate. So this world is going to have an interest rate in it. It’s going to have inflation in it. With me so far?

Let me add one more thing. I could come back to this. So I said that if you take thetaalpha less than theta-baralpha it means you’re selling the stock. So I’m going to allow people to go even further, thetaalpha less than 0. So let me just write that again, thetaalpha less than theta-baralpha means selling alpha. Thetaalpha less than 0 is doing a lot more than selling. You don’t have it to begin with, so what are you selling?

Well, the mathematics is telling you that over there thetaalpha’s going to be negative. Instead of getting extra dividends you’re going to be giving up dividends because it’s going to reduce your supply of money. So thetaalpha less than 0 is called selling short–I don’t know which one I’ve lost, but it’s got to be bad–if you can still hear me–you’re selling short, this means. So you’re selling something you don’t even have. It’s also called naked selling. It’s also called making a promise without collateral.

So I’m going to, for now, allow for that. So we’re not taking into account that anybody’s defaulting. If you take thetaalphanegative it means your income in the future is going to be reduced because you’re going to have to deliver the dividend because you’re going to have negative dividends, which means effectively you take out of your endowment those dividends and hand them over. So it’s as if you always keep your promises. So this model so far, the Fisher model, assumes no default, no collateral.

We’re not worrying about any of that stuff, and of course that’s going to be a critical thing. So you see something’s happened that we never had happen before. In the past you traded money for a football ticket. You gave up something you wanted you got something that you also wanted. It was a trade of value for value. Everybody agreed the two things you traded were equally valuable. If you take thetaalpha negative, by taking thetaalpha negative that becomes a negative number here so it allows you to spend more. You can buy more goods by taking thetaalpha negative. That’s negative. That means this can be more positive and still satisfy this constraint.

So by selling a stock short you’re promising to do something in the future. You get more money now you can eat more now, and then of course you have to consume less in the future because you have to pay back your promise. So you’re exchanging something valuable, you’re getting money, something valuable in exchange for a promise which is worth nothing until the future when you deliver on your promise. When you buy the stock you’re buying part of the tree, but the tree’s doing nothing for you now. You’re doing it because it’s going to be valuable next period. You’re actually not physically owning the tree, you’re owning a piece of paper that gives you a right to half the dividends of the tree. So you’re getting something that’s only good because it’s a promise you think is being kept.

Chapter 5. Present Value Prices [00:43:35]

So, so far for the next few lectures we’re going to ignore the fact that people get very nervous when they give something up that’s valuable in exchange for something that’s just a promise. So a critical thing has happened here. So we’ve kept the same mathematics except we’ve surreptitiously added this huge assumption. Now Fisher said, “Having done that, what can you realize?” This is the most important insight. He said this model it looks so complicated. It looks like now we have vastly more equations. No wonder Marx and all those religious zealots were getting confused. We can simplify it all and be back to where we were before and yet talk about finance.

So Fisher introduced the idea of present value prices. So he said look, when you buy a stock what are you really doing? This is the principle of arbitrage. He says when you buy a stock you’re saying to yourself, I’m giving up money today. Now, money today is consumption because I would have used that money. If I didn’t buy stocks I would have bought apples today and eaten them. So when I buy a stock I’m giving up apples today. I’m getting the stock which is then paying me dividends tomorrow, which whatever they are I’m selling off, I’m getting a profit out of the stock tomorrow and I’m ending up with apples tomorrow. Maybe I’m just eating the dividends straight off the tree. So when I buy a stock I’m really giving up apples today and getting apples tomorrow.

And no matter how I do it, whether it’s through stock alpha, or through stock beta, or through a nominal bond it’s got to be the case that all three ways, or all 50 other ways you could imagine doing it have to give me the same tradeoff. This is your yield you were talking about, the same tradeoff. The amount of apples I effectively give up today in order to get apples tomorrow is going to be the same no matter which way I do it. If it weren’t the same, if alpha’s price was more than a half of beta’s nobody would buy alpha. In fact they would start selling alpha. So that’s why this assumption is so important. What would they do? Not only would nobody would buy alpha, but they start selling it.

They’d say well, alpha is so expensive, let’s say it’s the same price as beta, I can sell alpha. With every alpha I sell I can buy stock beta, and so I haven’t done anything today, but in the future I’ve got stock beta which is paying me 2. I owe, because I sold stock alpha short I owe 1, so I’ll pay off the 1 I owe and I’ll still be left with 1. I’m making an arbitrage profit, and so I’m not going to stop at selling 1 share of alpha. I’ll sell 2 shares of alpha, then 3 shares of alpha, then a million shares of alpha, and everybody’ll be selling alpha short to buy beta and the market for alpha will never clear. So that’s why the prices will have to adjust.

And so it has to be in equilibrium, the price of alpha’s exactly half the price of beta, which is to say, in short, that if you solve for this equilibrium you can solve for an equilibrium where P1 = q1 is the price today of an apple today, and P2 is the price today (that’s why it’s called present value price) of an apple next year. So if you’ve got this equilibrium by working your way through, by figuring out what the price of alpha is–so the stock, for example, you want to figure out what the stock is. Suppose the stock of alpha, suppose the price turns out to be a half.

Then by paying a half today you can buy stock alpha, which is going to pay you a whole dividend. So the price, therefore, of an entire–oh, let’s do beta. Suppose the price of beta is a quarter. Suppose we happen to find out that the price of beta is a quarter. Then what’s P2? How much do you have to give up today in order to get an apple? Well, by paying a quarter today, that’s the price, by paying a quarter today you’re getting 2 dividends. So by paying a quarter today you’re getting 2 dividends. If you paid to get 1 dividend you’d have to pay an eighth today. So the price P2 would be an eighth in that case. So by piercing through the veil of the stock market you can always figure out what you’re effectively paying today in order to get an apple next period.

And that price which was just computed would be the same whether we looked at it from the point of view of going through stock beta, or through stock alpha, or through the nominal bond. It would always have to give us the same answer. So we know, from the financial equilibrium, we can deduce what P1 and P2 have to be, the present value prices. And so effectively, furthermore, stocks effectively just add to the endowments of goods. So we can now consider another economy. So let’s consider the economy E-hat, so the hat economy. So U-hat-A of X1 and X2 is the same as it was before, UA of X1, X2, U-hatof X1 and X2 is the same as it was before, but endowments now E-hatA1 E-hatA2 is going to be what?

Well, A over here began with 1 unit of each good, but A also owned all of stock alpha and half of stock beta. So all of stock alpha pays 1 dividend in the future, so really A effectively has claim on two apples in the future and another half of beta which is another apple in the future, so really A’s initial endowment of goods is (1, 3).

How did I get that again? I said it was 1 apple to begin with he could anticipate having. He knew he owned all of stock alpha which pays 1 apple, so that’s another one that’s really his, and then in the future he’s going to get half of the dividends of stock beta, and half of 2 is also 1. So he’s got 3 apples in the future. And E-hatB2, well his 1 doesn’t change today, but what’s his claim, effectively, on dividends in the future?

Student: 1.

Professor John Geanakoplos: 1, thank you. Somebody answered that. So we’ve now reduced the financial equilibrium to a general equilibrium, the same kind of economy we had before. It’s just that we had to augment the endowments to take into account that people own stuff through the stocks. So what’s the equilibrium of this economy?

This has a simple general equilibrium. So what is it? How do we solve for equilibrium? Well, take P1 = 1 and we’ll solve for P2. So let’s just clear the first market. How do you clear it? You’re with me here? It’s a standard general equilibrium, the same kind we’ve done many times before. So see if I can do it. So person 1 is going to spend a third of his money [correction: will be two thirds], and how much money does he have? He has (1 + P2 times 3), that’s A, right? His endowment is (1, 3). This is his income, and he’s spending a third of it on good 1. And the price of good 1, P1, here is just 1. And then B is going to spend, he’s a half, half Cobb-Douglas guy, so this is 1 and this is 1. He’s spending half of his money and his income is [1 + P2 times 1] divided by 1, and that has to equal the total endowment which is 2, 1 + 1. So did I go too fast? Yes?

Student: Why is it a third?

Professor John Geanakoplos: Well, it’s probably wrong. So let’s try it again. Maybe it’s 2 thirds. Let’s see what I was doing. I’ve taken the financial economy, which was very complicated, looks very hard to solve and Fisher says of course when we add uncertainty and things like that we’re going to have to do other tricks. But without uncertainty, with perfect foresight and so on, and no uncertainty, Fisher says this is an easy problem to solve. You take the financial equilibrium with all its extra variables and you realize if people are rational they’re going to see through all that complicated stuff.

They’re going to realize that alpha is just half as good as beta, and so they’re going to realize that by holding stock they’re making a certain tradeoff between alpha and beta. And we calculated the tradeoff. What was P2? I forgot what P2was. Anyway, how much did you have to pay? If you pay pialpha divided by Dalpha2, something like that, was P2. So if it costs you a certain amount of money, if it costs you a quarter we said, so this is P2, so through either stock, like beta’s the one I solved it for. I said suppose beta, that’s also equal to pialpha over Dalpha2, we said if the price of beta turns out to be a quarter and you’re getting two dividends then by paying a quarter you get two dividends. So it means to get one apple it only costs you an eighth, an eighth of a dollar. So P2 we can figure out. So once we’ve got our financial equilibrium, it basically is determining a general equilibrium. So instead so let’s go backwards.

Instead of solving for the financial equilibrium that looks complicated let’s solve for the general equilibrium. What is the effective general equilibrium? It’s the same utilities as before, but we’ve augmented the endowments. By looking through the veil of the stocks we realized that A actually owns 3 apples in the future, 1 because he owns all of stock alpha and another one because he owns half of stock beta. So we’ve got this simple economy that we’re used to solving that you did on the first problem set, so we can do it again and solve it. So I’m going to now solve for general equilibrium. I have to solve for P1 and P2 and all the XA1, XA2, XB1, XB2, but I can fix one of the prices to be one. So I’ll fix P1 to be 1.

Then what does A do? So I made a mistake which is why you weren’t following me. A, his Cobb-Douglas, 2 thirds of the weight is on good 1 and 1 third on good 2. So he’s going to spend 2 thirds of his money on the first good. So that’s why this should have been a 2 thirds as she pointed out, thank you. So 2 thirds of his money, what’s his money? His endowment is (1, 3), so it’s [(P1 times 1), which is 1 times 1, + (3 times P2)] divided by the price P1. 2 thirds of the income divided by the price of the first good, that’s how many of the first good he wants to eat.

What does she want to do? She’s patient. She’s going to spend half her income on both goods, so half of her income which is [(1 + P2 times 1) divided by 1], that’s how many apples today she wants and that’s what we have to clear to clear the apple market at time 1. So does this make sense now? I’m looking at you in the front. Do you agree with this or is this confusing now? Do you follow this or is this confusing? I can say it again if it’s confusing.

Student: I’ve got a question.

Professor John Geanakoplos: Yeah?

Student: Our denominator represents what they want to have in the future?

Professor John Geanakoplos: Remember how the Cobb-Douglas worked? This trick I’m going to use over and over again. With log utilities everybody will spend depending on the coefficient. So remember, this utility in the problem set, you know that this utility is just the same as if I put 2 thirds here and 1 third here, right? Because I’m just dividing this by 3, so instead of 1 and a half–so the original utility is this, right? It’s (log X1) + (1 half log X2), so the sum of this plus this is 3 halves. I can multiply by 2 thirds. So if I multiply this by 2 thirds I get 2 thirds here and 1 third here, right? And I haven’t changed the utility function.

And now I know this is a familiar pattern. A is always going to spend 2 thirds of his money on the first good. And B I can multiple this whole utility by a half and a half and B is going spend–and now we recognize it as the common Cobb-Douglas thing and we could say that A’s [correction: B’s] going to spend half his money on the first good and half his money on the second good.

Chapter 6. Real and Nominal Interest Rates [00:57:44]

So what have I done? Fisher said look, this model is so complicated. You’re thinking in your heads, people are deciding in period 1 how much stock should I buy, how many bonds should I buy, how many apples should I eat, but really if they’re smart they’re not going to think that way. They’re going to say to themselves how many apples should I buy today? How many apples do I want to consume tomorrow? All these financial assets are just methods for me getting apples tomorrow in exchange for apples today. And what’s the tradeoff between the apples was this–P1 and P2 is the tradeoff between apples.

You can look through the stocks and all that, but no matter which stock you think of buying there’s going to be the same tradeoff between apples today and apples tomorrow because of the no arbitrage. The price of alpha is going to have to be exactly half the price of beta. So once I solve for this economy and get the price of alpha I’ll know how many apples today I have to tradeoff in order to get apples tomorrow. So I might as well forget about all the stocks and just try to figure out what must that tradeoff between P1 and P2 be.

So that’s why you can forget about the stocks, forget about the bonds, everybody’s thinking I’m trading off apples today for apples next year. I’m making all the trades today, because I’m trading apples today in exchange for promises for apples next period. So it’s as if everything happens today. It’s as if they’re present value prices today. We trade today at prices P1 and P2 for apples today and apples next year.

Of course the apples won’t appear until next year, but I can sell an apple today at price P1 and buy promises for apple next year at a price P2, and that’s the tradeoff I’m facing. If I face that tradeoff how much of my money am I going to spend on apples today? I’m going to spend 2 thirds of my money on apples today and the other third I’ll spend on promises for apples next period. So this is a big insight Fisher had. It’s not surprising it’s a little puzzling. I’m so used to it that I’ve forgotten how puzzling it is. So ask me some more questions. This was not an obvious thought Fisher had. Yeah?

Student: Do we typically expect the price in the 2nd period to be lower than the price in the 1st period?

Professor John Geanakoplos: Often there’ll be inflation, so q2–the contemporaneous price next period might be higher than the contemporaneous price this period, but we don’t care about that. What we care about is how many apples you have to give up today in order to get apples tomorrow. So P2 is the present value price. What do you have to give up today to get the apple next period? So we expect P2 to be less than P1. Precisely because, well, we’re going to come to that, that’s the next thing I was going to talk about because everybody’s putting more weight on consumption today than they are on consumption in the future. That’s why the price P1 is going to be bigger than the price P2. Yep?

Student: When we’re solving it we’re solving in real prices?

Professor John Geanakoplos: So we’re solving for P1 and P2 in present value prices. So the crucial thing is, he invented this term present value prices: the prices you pay today no matter when you’re going to get the stuff. That’s his big insight. You should look at present value prices. Holding stocks and all that complicated stuff is just giving you goods in the future. So when you buy the stocks today you should think, how much am I having to pay today to get an apple in the future?

You can deduce that from the price of stocks and how many dividends they’re paying. So everybody must have figured out a P2. What does it cost today? How much money do I have to give up today to get an apple in the future? Well, I have to buy a stock and then sell the dividends and all that, but really what I should be thinking about is what’s the price today I’m paying for one apple in the future, and that’s P2. And so when you think about it that way, although it’s an intertemporal problem, it looks like a new model with time, Fisher said you can reduce it, think of it as if they’re just the same problem we did before with two goods you’re trading at the same time. That’s not an obvious thing to have thought of. No one thought of it before him. Yeah?

Student: The budget set, the second equation from the right-hand side. The second item shouldn’t you have a q2 as well, q2 times <>?

Professor John Geanakoplos: Oh, absolutely. So this should have been a q2 here because you’d sell the dividend and you get money by selling the dividend. Thank you. Ho, ho, ho, very good. Who said that? Who just asked that question? Where are you? I’ll remember you. That was very good, exactly.

So in the future you’re getting the money. So what he pointed out is I made another mistake here. In the future the money you’re spending on goods in the future you’re going to get the dividends paid, of course you can sell the dividends for money and the price is q2. So a q2 has to appear over here just like there’s a q2 over there. So it’s the goods times the price. That’s the money you’re getting in the future, and that’s the money you’re spending on the good X2. Very good, too bad you didn’t ask me that a while ago, but anyway. Any other questions? So we’re back to this standard general equilibrium problem. We can take a financial equilibrium and turn it into a general equilibrium. And so when we solve this we’re going to have (2 thirds + P2 + 1 half + 1 half P2) = 2. So it looks like 3 halves (I hope I haven’t done this wrong) P2 = 2 thirds + 2P2.

Student: <>

Professor John Geanakoplos: Thank you, yeah, plus 2P2. So we have 2P2 + a half + a half P2 so we have 5 halves P2. That was lucky you caught that, 5 halves P2. So 2 thirds is 4 sixths. And 3 sixths is 9 sixths. And 12 sixths - 9 sixths is… What is this? So what’s 2 - 2 thirds - a half?

Student: 5 sixths.

Professor John Geanakoplos: 5 sixths. That’s correct. So P2 therefore equals 1 third. All right, so we’ve now solved for equilibrium. We know that P1 has got to be 1. P2 has got to be 1 third. We know that we can figure out what consumption’s going to be, I mean XA1, for example, if we wanted to solve for that we just plug in a third here. So we’d have (2 thirds) times (1 + 3), which is 2 thirds times a 4, which is 8 thirds, I guess.

And XB1, we could have solved for that too if we wanted to. XB1 is going to be a half times 4 thirds which is 2 thirds. No, that doesn’t–a half plus what was this 1 third? No, it’s 1 + 1 third which is 4 thirds times a half which is 2 thirds. Is that right? That doesn’t look right, so maybe I did this wrong. 1 + 1 is 2 so this is 4 thirds. That looks better. XA1 is 4 thirds. XB1is 2 thirds. So we have 4 thirds and 2 thirds, and so we could solve similarly for XA2 and XB2, which I won’t bother to do. So we can figure out what the prices are, the present value prices and the present value consumption.

But having done that, Fisher says, we took a hard problem we make it easy. Let’s go back to the hard problem. So Fisher says the tradeoff between good 1 and good 2 is 1 to a third, so he defined–here’s the nominal rate of interest. Fisher defined something called the real rate of interest. And he said that was a variable that you should pay a lot of attention to. So the real rate of interest he said is P1 divided by P2, so this is equal to 3 and so r is 200 percent. So how did I get that? Just as someone in the front said the good 2 is much less expensive. The present value of good 2 is much less than the present value of good 1. People think an apple today is much more valuable than an apple tomorrow. So if you give up an apple today you can get 3 apples next year.

So if you put an apple in the bank it’s like getting 200 percent interest on apples. So he called that the real rate–the apple rate of interest. You put an apple in the bank, you give up an apple today, buy stocks and when it comes out in the end you’ve got 200 percent more apples than you started with. So it’s the real rate of interest. So that’s his crucial variable. Now, let’s go back to the original equilibrium. What is the stock price? Assume q1 = 1. What is the stock price pialpha? Well, we can figure it out. How can we figure it out? What is pialpha? Well, stock alpha pays 1 good tomorrow so what is the price of pialpha?

Student: <>

Professor John Geanakoplos: What? Somebody said it. I couldn’t hear it–a third. How did I get a third? Because we figured out that once everybody looks through the veil, assuming the price P1 is 1 and the price q1 is 1, if they look through the veil they’re going to say to themselves ah-ha, how much do I have to pay today to get an apple in the future? I have to pay a third to get one apple in the future. P1 is 3 times P2. So to get 1 apple in the future, it’s only a third of an apple today. So the stock pays 1 apple in the future so therefore how much do I have to pay today? I have to pay 1 third of an apple today, and since I took the price of apples to be 1 it’s going to be the price of 1 third. So what’s the price beta?

Student: 2 thirds.

Professor John Geanakoplos: 2 thirds. So Fisher said look, we’ve solved now for all these financial things. So what you can’t do, Fisher’s theory does not explain how much of each stock, thetaA, etcetera, the investors hold. Why is that?

Well, because it doesn’t matter. Not enough is happening in the economy yet. Alpha and beta are exactly the same. If you own twice as much of the alpha tree you get exactly the same as having the beta tree. So how can you possibly tell whether somebody’s going to hold twice the alpha tree or just one beta tree? Either way he’s going to get the same thing. So the theory can’t possibly explain which one they’re going to do. Somehow they’ll work it out and divide up the tree so that everybody ends up with the right number of apples in the end.

And it also does not explain inflation because you can’t tell what q2 is going to be. All right, because you see in this budget set, thanks to that inspired question, if you double q2, q2 appears everywhere; you’re not going to change the second equation. So q2 could be anything. You can double it or triple it, it won’t matter, and the same with q1. It’s just like Walras said before, you can always normalize prices to be one. He had to add another theory of money and how many dollars were floating around in the economy to explain q2. This theory won’t explain it.

So it does not explain inflation, and it does not explain who holds which stock, and so it does not explain the nominal rate of interest. It does not explain i, the nominal rate of interest, because 1 dollar, who knows what 1 dollar’s going to be worth. It depends on how much inflation we have.

But it does explain the real rate of interest. It does explain r, and that’s the variable that Fisher said is the one economists should always pay attention to, the real rate of interest. So that’s the crucial variable. So if you want to figure out what’s the price today of a stock, so Fisher’s famous equation’s the price of the stock today, so pialpha divided by q1, so the real price, as somebody said, the price in terms of goods of the stock today is always going to equal the dividend in the future divided by 1 + r.

Why is that? That’s exactly what we already used. This is just a rewriting of the trick we did before. You take the dividend tomorrow, you multiply it by P2 and then you realize that 1 + r is just P1 divided by P2, so replacing the P1s and P2s by q1and 1 + r, today’s real stock price is just the dividends tomorrow discounted. This is what he called the Fundamental Theorem of Asset Pricing. If you knew the real rate of interest you’d be able to figure out what all the stocks were worth just like we did. Once we knew P1 and P2, the present value–the present value prices determine the interest rate because they’re–just as we said. P1 over P2, remember, is 1 + r. So knowing P1 and P2 you’re always normalizing P1 to be 1, P2’s the same as 1 over 1 + r. So if you know P2 or you know 1 over 1 + r, you know what the value of the stocks are. That’s his critical insight. Now, just to finish… Yes?

Student: That’s q2 or q1?

Professor John Geanakoplos: q1, q1 which is the same as P1 because it’s today’s price. The contemporaneous price today is the present value of the price today. So let me just end on this one note. Fisher said we can take financial equilibrium without uncertainty, reduce it to general equilibrium. We know everything about general equilibrium, therefore we know everything about financial equilibrium, and we realize that the crucial variable in general equilibrium is relative prices.

There is no just interest rate. The nominal interest rate, who the hell cares? The real interest rate is what we care about, and just in normal economics there’s no just relative price, there’s no just real interest rate. It depends on people’s utilities. You make them more patient and that’s going to affect the real interest rate. You make them less patient it’s going to affect the real interest rate. You give them more endowments today versus tomorrow, that’s going to affect the real interest rate. The relative price between today and tomorrow–that’s the way you should think about finance. That’s the way you should explain what’s going on in the financial markets. So in the problem set you’re just going to do a problem like that, and then I’m going to give more interpretations of this that Fisher gave. So I guess I’m out of time, so we’ll stop here.

[end of transcript]

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