ECON 251: Financial Theory
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ECON 251 - Lecture 6 - Irving Fisher's Impatience Theory of Interest
Chapter 1. From Financial to General Equilbrium [00:00:00]
Professor John Geanakoplos: All right, so we spent a long time reviewing general equilibrium and we’ve now switched to finance, and you’re hopefully going to see that the principles of finance emerge very quickly from the principles of general equilibrium. So that although it seems it was a long interlude we’ve actually learned a lot about the financial economy. So I’m going to continue with the example that we started with the last time. So we have a financial economy. So in a financial economy–what is a financial economy?
On this top board the financial economy is defined by lots of people in the economy and their utilities. So here we have for simplicity two kinds of people A and B with utilities given by the log X1 + 1 half log X2 etcetera. It’s also people know today what their endowments are and they have some idea of what they’re going to be tomorrow. They’re labor powered today and they’re going to be able to work again next year. So the labor endowments are given by (1, 1) for A, and (1, 0) for B.
And then they also know that there are two stocks in the economy and they have to anticipate what the dividends are going to be. And as Fisher said, the main value of assets is that they give you something, they produce something. In this case they’re going to be dividends and beta’s producing dividends of 2, and alpha is producing a dividend of 1 next period, and then the ownership of shares.
So that’s the beginning of the economy and we want to define from that equilibrium which involves: what are the contemporaneous prices going to be, that’s Q for contemporaneous, what are the prices of the stocks going to be, and who’s going to hold which portfolio of assets of stocks, and who’s going to consume what. And so Fisher said that’s a very complicated problem. You can simplify it by looking at a general equilibrium problem which is much shorter to describe. And so the general equilibrium economy is going to be a much simpler one.
It’s going to consist of UA and UB the same as before, and E-hatA1, the endowments, E-hatA2 and (E-hatB1 E-hatB2). So we’ve left out half the variables up there and we define E-hatA1 = EA1 = 1 and E-hatB1 = EB1 = 1, but E-hatA2 (this is the Fisher insight) = EA2 + what A owns of the payoffs of the future dividends, [theta-barAalpha times Dalpha2 plus theta-barAbeta times] Dbeta2. Since A owns half of the alpha stock, sorry, all of the alpha stock and half of the beta stock, his endowment is 1, his original thing, plus what the stock is going to produce, and after all he’s the owner. So he’s going to get all of 1 + a half of 2 which is = 3. I took more space than I thought. And so similarly E-hatB2 is going to be 1 + a half of 2 which = 2. So here endowments are this and also let’s just write it here, E-hatA2 = 3, so this.
So Fisher said we start with a financial equilibrium, we can switch to the economic equilibrium and solve this problem, and having solved that one go back and figure out how to solve this one. And you remember what the prices were. They turned out to be q1–I might as well write it up there what the prices we had, we solved. We said first of all Fisher has no theory for the contemporaneous prices. It’s all relative prices. I’m going to write that. Relative prices, is all we can ever figure out. Someone might always come along and change dollars to cents.
When I was a little boy in France, on vacation, they suddenly announced that the franc was going to be divided–everything that was a hundred francs would now be one franc. They just redefined the currency, so that might always happen. So you have to have some theory of money and whether the government’s going to do that to figure out the nominal prices. So contemporaneous prices he says are 1, 1.
All right, but having realized that if there are many goods at time 1 he could figure out the relative prices, but with only 1 good at time one who’s to say whether we’re measuring dollars or francs or cents, we’ll just call it 1, and the same thing’s going to happen next year. Who knows whether it’s dollars or cents or francs so we’re going to call it 1 again.
But after that he figured everything out. This turned out to be a price of a third, this turned out to be a price of 2 thirds and we figured out all the consumptions, which I’ve forgotten, of course. But anyway they were–who knows what they were, not that it’s too important. All right, well I forgot what they were. Anyway, he figured out all the consumptions. I think they were–actually I sort of remember them. Well, let’s say I don’t. Anyway, he figured out all the consumptions. Does anyone remember what they were?
All right, I will look them up, 4 thirds 2, 2 thirds 2, so they were 4 thirds and 2, and 2 thirds and 2. He figured out in equilibrium, and how did he do it–because he solved over here first. We would have solved–he didn’t do this exact problem, but he would have solved over here and we would have found with P1 = 1, P2 = a third, and sure enough XA1 = 4 thirds, XA2 = 2, and XB1 = 2 thirds, and XB2 = 2.
So Fisher said start with the financial economy, figure out what the reduced general equilibrium is, solve for this equilibrium, and go back and figure out what the financial equilibrium should be.
Chapter 2. Applying the Principle of No Arbitrage [00:06:44]
All right, so I want to now examine what we’ve done. And we did that the end of last class. You had to do it in a problem set. And you notice that the only difference between this and that is, the general equilibrium throws away a lot of irrelevant information because Fisher said people are rational. They look through the veil of all the gibberish of who owns the company and stuff like that, and they’re just anticipating what the company is going to produce. They don’t really care about whether there’s a man running the company, or a woman running the company, or whether she’s got an MBA from Harvard or from Yale. None of this is relevant, what the business plan is. All they care about is what’s going to actually happen in the end. So if you think they’re going to anticipate that correctly you don’t need to worry about all the other stuff. So looking through the veil you can always reduce the financial equilibrium to a general equilibrium.
Now, I want to go back and reexamine all that logic. So what’s the first step in what Fisher did? And this is the idea of no arbitrage. So Fisher said people look through the veil of things. They understand stuff and you can count on their understanding to guide your understanding of the economy. So if you know that pialpha–(this is a big pi)–pialpha = a third, so Fisher says well, you don’t have to solve for the whole equilibrium to figure out what pibeta is. What would pibetabe?
Well, Fisher would have said stock beta always pays off exactly what stock alpha pays off. So if these people are rational they’re not going to allow for an arbitrage. So arbitrage means if there are two assets or two things that are identical, they have to sell for the same price–that’s no arbitrage. If they sold for a different price there’d be an arbitrage. You’d sell the more expensive one and buy the cheaper one, and so you’d have accomplished a perfect tradeoff, but you’d have gotten the difference of money. So since pialpha is 1 third, pibeta has to equal 2 thirds.
That’s the first, most important principle of finance that Fisher introduced; the idea of no arbitrage and making deductions for no arbitrage, so most of finance is actually being more and more clever about how to do no arbitrage. Over half of this course is going to be, let’s look at situations where at first glance there doesn’t seem to be any arbitrage. Then you realize if you’re clever enough you’ll recognize an arbitrage and be able to figure out all the prices without having to know all the utilities and everything else–so one of the main goals of finance is to explain asset prices. You can see how no arbitrage is going to help do that, because if you knew what some of the asset prices were you could deduce what the rest might be. So that’s the first thing Fisher did, and he’s used this fact in connecting these two economies.
So that’s the first thing. Now, that principle can be used over and over again. Another application of it, let’s suppose that we introduced a nominal bond with payoff 1 dollar in period 2. And suppose, as before, that q1 = q2 = 1, as we’ve already supposed. So then by definition the price of this bond is equal to 1 over 1 + i, where i is the nominal interest rate. Why is that? Because you’re going to get a dollar next year. If the price is less than a dollar this year you’re turning something less than a dollar into something equal to a dollar. You’re multiplying today’s price by 1 + i to get tomorrows price, so the rate of return is 1 + i, taking whatever you put in today and getting 1 + i tomorrow. So what is 1 + i?
So by no arbitrage we can figure out what 1 + i must be. So 1 dollar today can go into 3 units of stock alpha, which goes into 3 units of X2 as dividends, which equals 3 dollars. So you take 1 dollar today by buying stock alpha you can get 3 units of it since its price is a third, and since stock alpha pays one unit of output next period you know that 1 dollar today gives you 3 units of stock alpha, which gives you 3 units of good 2 as the output and at price 1 dollar tomorrow you’ve anticipated that’s 3 dollars. So by buying stock alpha you can put in a dollar and get out 3 dollars. So it means that 1 + i = 3, which means the interest rate is 200 percent. So that’s a second thing you can deduce from that.
So notice that by looking at part of the equilibrium here we can figure out a lot of the rest of the equilibrium. So what’s another application? Well, Fisher said define the real interest rate as number of goods today goes into number of good tomorrow. So this will be, 1 + r equals that. The number of goods today and how many good tomorrow do you get? So how can you do that? Well, 1 good today, 1 unit of X1 is 1 dollar today, right? If you had one apple today you could sell it for q1 times 1 apple, which is q1 times 1, which is 1 times 1, which is 1 dollar today, which you can get 3 units, 3 shares, 3 units of stock alpha, which gives you 3 units of X2. So 1 unit of X1 today turns into 3 units of X2, so therefore 1 + r = 3 implies r = 200 percent. So that’s the real rate of interest. So one of the tricks in going from here to here was to say that Fisher realized that people are going to look through all the gibberish of money and they’re going to think about what apples are they giving up today and what apples are they getting tomorrow. They’re not going to be confused by all the holding of assets in between.
All right, so let’s just make it a little bit more complicated. Suppose we started with q1 = 1, q2 = 2. Now, I told you that equilibrium–Fisher says there’s always a normalization. Walras originally had the normalization in one period. There’s a one period model in general equilibrium. In multi-period models there’s a normalization every period. Every period there’s a choice of whether you’re dealing with dollars, or francs, or centimes, or how many, and so there’s a free normalization. So let’s take q1 = 1 and q2 = 2.
Well, what does that mean? That means that inflation 1 + (let’s call it growth of money) g–i, I’ve already used for the nominal interest rate. So, 1 + g is going to be 2 over 1 or just 2. So inflation = 100 percent. So what’s pialpha going to be? I’ve done my work. Now the rest I’m going to just ask you for the rest of the numbers. What’s pialpha? So if I re-solved equilibrium taking q1 = 1 and q2 = 2 all that’s kind of money stuff so it’s not going to change what happens over there. You’re going to get the same equilibrium over there and you’re going to go back to over here. So what’s pialpha going to be?
Ah-ha! Suppose we knew we were in the same real economy. There’s nothing changed about utilities, endowments of goods, productivity of the stocks. All we know is that inflation’s going to be higher now. So what do you think would happen in the new equilibrium? What’s going to happen to the price of stocks today? Yes?
Student: Is it just 2 dollars?
Professor John Geanakoplos: Price of stock alpha. What was it before?
Professor John Geanakoplos: So what was it before?
Student: 1 third.
Professor John Geanakoplos: Yeah, 1 third, so it’s still 1 third. This is a big mystery in finance, a big question in finance. So you see why it’s puzzling. You didn’t get the answer right off, although she did. So you just have to think about it a second.
If you really thought that people when they were buying and selling only bought a stock because they said to themselves, “How many apples am I going to get out of this stock? I don’t care about dollars and centimes and francs. I’m not going to eat that. I’m going to eat the apples, and maybe I get the apples and sell them and eat pears instead, but I care about the goods I’m going to get. So I looked through all the veil.”
I should recognize that the stock, although it’s now going to pay twice as many dollars as it did before, so it’s going to pay 2 dollars. That’s how someone guessed 2. Someone said 2. So how did he get 2? I didn’t even realize how he came up with the number 2. He came up with the number 2 because he said, well the stock is paying 1 apple tomorrow, the price of apples is now 2, so it’s paying 2 dollars tomorrow so maybe its price today should be 2. But no, that isn’t how much the stock is worth.
The stock is worth solving for this general equilibrium supply and demand. We already calculated before that the stock was a third, so the price of the stock is going to stay a third because the apples it pays tomorrow hasn’t changed. It’s still the same one apple. Now, how did we know the stock was priced at a third before? What was the stock in general? What’s the price of the stock?
The price of stock, remember, is how did we get it by going from here to here? We said it’s going to equal the price of the stock divided by P1. Now, the stock is only paying a certain number of goods. The price of the stock today is going to equal the present value 1 over (1 + r) times its dividend. I’ll write it this way. The price of the stock is P2 times this. Let’s just write this.
What would Fisher say? How did we get the price of the stock from going from here to here? We got the price of the stock by saying the stock pays off one good tomorrow, but one good tomorrow is only worth a third of one good today, so therefore the value of the stock is only equal to a third times 1 = 1 third. So assuming P1 = 1 that’s what Fisher would say. Assuming P1 is 1 you figure out how many units of today’s good is it worth.
Now, if P1 isn’t 1 then what do you do? Suppose P1 were 5 and P2 were–or P1 is 6, let’s say, and P2 is 2 then what would you do? You’d have to say P1 times pialpha = Dalpha2. So if you multiplied all the prices by–am I putting the P1down at the bottom or the top? If you multiply out all the prices by 3, just leave it like this. We’ll say if pialpha = P2 times Dalpha2. If you measure it in terms of goods, that’s how you do it.
So if you take this, this is also equal to 1 over P1 divided by P2 (if P1 is 1, assuming P1 is 1) times Dalpha2, which is 1 over (1 + r) times Dalpha2. So Fisher said–so here’s his famous equation. Fisher said the way to figure out the value of a stock, if you solve that problem over here, is to look at its dividends and discount them by the real rate of interest–1 unit of output tomorrow, since the value of an apple tomorrow is only a third of the value of an output today. Remember the interest rate 1 + r, the real interest rate, is equal to the ratio of the two goods. So P1 over P2 is just 1 + r, 1 + r is P1 over P2.
I’m making some things simple seem more complicated, sorry. So let’s just say it again. When we solved that equilibrium over there we figured out that P2 is only a third of P1. When people think today how much would I give up of apples today to get an apple next year they don’t think apples next year are worth nearly as much as apples this year. So they’d only give up a third of an apple this year to get an apple next year. P2 is the amount you give up today to get an apple next year, so it’s a third.
Another way of saying that, if P1 is 1, is that the real interest rate, the tradeoff between apples tomorrow and apples today which is P1 over P2, because 1 apple today can give you three apples tomorrow, so P1 over P2 is 3, so 1 + r is three. So the apple tomorrow is worth P2 times the dividend. That’s just 1 over (1 + r) times the dividend.
So the value of a stock is the real dividends it’s paying in the future discounted by the real rate of interest. You’re turning tomorrow’s next year’s goods, finding the equivalent in terms of this year’s goods, and the ratio of those two prices is the real rate of interest and so that’s how you would get it.
Chapter 3. The Fundamental Theorem of Asset Pricing [00:23:50]
So another way of saying the same thing is you could turn cash next year into cash this year. So assuming q1 is 1, another way of saying that is 1 over 1 + i times Dalpha2 times q2. So you take the nominal rate of interest times the money that’s being produced, because the nominal rate of interest says how do you trade off a dollar today for a dollar in the future? So a dollar in the future isn’t worth, usually, as much as a dollar today so you have to discount it. So a certain number of dollars in the future are worth less dollars today. So you take the payoff of dollars in the future discounted by the nominal rate of interest and you get today’s price, or you take the real dividends in the future discounted by the real rate of interest and you get today’s price. So both those things are an application of the principle of no arbitrage, looking through the veil.
So what would the nominal interest rate be in this case? In this case you see, how did I know that Palpha was still a third? Because the real interest rate hasn’t changed, it’s still 200 percent. So Dalpha2 is 1 and I’m still multiplying by 1 third, so I’m still getting a third for the price of alpha. So that’s how she knew that the answer should stay a third because she knew nothing real had changed in the economy, therefore the real interest rate couldn’t have changed, therefore the price of the stock still had to be the same.
So how could we have used this [clarification: another formula] formula? We have to know what the nominal interest rate is. So what is the nominal interest rate? If you put in a dollar today how many dollars can you get out in the future in this new economy where there’s 100 percent inflation? Yes?
Student: 500 percent inflation.
Professor John Geanakoplos: So that’s right, now how did he do that? Because let’s just do it. You take 1 dollar today at price q1 = 1. You can buy 3 units of alpha still, because its price is still a third, and that tells you that you get 3 units of X2, that’s the dividend. Of 3 units of alpha each share of alpha pays 1, right, 1 apple, so now you get 3 apples, but that’s equal to 3 times 2 because the price is 2, = 6 dollars tomorrow.
So you’ve turned 1 dollar into 6 dollars. So 1 + i = 6 over 1 implies i = 500 percent, just exactly what he said. So to say that just more simply the real rate of interest 1 + r, this is the most famous equation Fisher ever wrote, is 1 + i divided by 1 + g. So this is called the Fisher Equation. His two famous equations are this, this is called the Fisher Equation and this which is called–these two things which are the same are called the Fundamental Theorem of Asset Pricing. So why is this theorem true?
The real rate of interest trades off apples today for apples tomorrow, the real rate of interest, apples today for apples tomorrow, so we had 1 apple giving you 3 apples. That’s why r was 200 percent. Well, if inflation is 100 percent, so this is 2, 1 apple today gives you 3 apples in the future, but that means 1 apple today gives you 1 dollar, is one apple today gives you 3 apples or 6 dollars in the future. So 3 times 2, so if this is equal to 3 and inflation’s 100 percent so this is equal to 2 then what’s the fair rate of interest? What will the banks give you?
Well, any banker can take a dollar, buy a stock, turn it into 3 units of dividends and then sell it for 2 dollars apiece and get 6 dollars. And so a banker can take a dollar and turn it to 6, so competition will force the bankers to give you 6 dollars for every 1 dollar you give it, next period. So the interest rate has to be 1 + i = 3 times 2, or 6. So the real rate of interest is the nominal rate of interest divided by inflation.
So that’s one subtle, but once you realize it, obvious implication of thinking people are rational and make sort of simple calculations looking at the future. And a consequence of that is the price of assets, or you look at the cash that comes out in the future discounted by the nominal interest rate, or you look at the real goods that come out in the future and discount it by the real interest rate, and it’s all the same thing.
So does anybody know what the inflation is today, or what the nominal interest rates are today? So i is the nominal interest rate, the amount of interest you put in the bank and what they’ll pay you at the end of a year. So we’re going to–next class we’re going to find out the exact numbers, but what do you think it is about? Does anyone have any idea? Take a wild guess. Is it 10 percent, 5 percent? Yep?
Student: I think the inflation is usually around 3 percent.
Professor John Geanakoplos: Usually, and do you think it’s higher or lower than usual now-a-days?
Student: It’s probably lower.
Professor John Geanakoplos: That’s good. So let’s say it’s around 2 percent. So that means this is 1.02 and what do you think the nominal interest rate is now?
Student: 1 percent.
Professor John Geanakoplos: Who said that? That’s a good–1 percent, that’s about right. So what is the real rate of interest now?
Professor John Geanakoplos: What?
Professor John Geanakoplos: Well, 1 + r is less than 1. So 1 + r is around .99. So the real rate of interest is actually like negative 1 percent. How did that happen? Do you think it’s standard to have the real interest rate be under 0? So why is it under 0? What’s going on now that would make that happen? Yep?
Student: The Federal Reserve wants to stimulate investment.
Professor John Geanakoplos: Ah ha! The Federal Reserve has cut the interest rate, the nominal interest rate that it lends at to close to 0, let’s say to 1 percent on the 1 year bond to 0 on the 3 month thing. So the reason they’re saying they’re doing that is to stimulate investment. That’s what they teach you in macro, Keynesian, stimulate investment. We’re going to find out that that’s not the reason they’re doing it at all. The reason the Federal Reserve is cutting the interest rate to almost zero is to just give money away to the banks, and why it that?
Well, when you put your money and deposit it in the bank you’re getting almost no interest, so the banks, the big banks have got all these deposits and people don’t change what they do. They just leave their money in the banks getting no interest. So the banks have the money for free and they can make money with it.
So normally they’d have to pay 3 percent interest or something and that would be expensive for them, and that expense is a big part of their expenses, they don’t have it anymore. So we’re going to come back to that what’s really going on today, but that’s what’s going on. But anyway, the point is the nominal interest rate is somehow controlled by the Fed. That’s why we don’t have a theory of it. We’re not going to do macro in this course. So Fisher doesn’t have a theory of the nominal interest rate, of inflation, but he does tell you, given inflation and the nominal interest rate, that’s determining a real interest rate, and people should look through that.
So now they should say, this is sort of the Keynesian part, they should realize that actually an apple today if you just sort of put it in the bank you get less than an apple in the future so you should spend it and do something with it. That’s the Keynesian idea, so people–why fritter away part of your apple, do something with it. That’s why it’s supposed to stimulate demand and activity today. So the point is, that’s how you calculate the real rate of interest and shockingly it’s negative and it’s hardly ever negative, but it can be negative.
Are there any questions about this no arbitrage business? All right, so let’s do one more trick here, a Fisher thing. So let’s go back to the equilibrium with q1 = 1 and q2 = 1. Suppose China offered to lend money, lend us dollars at a 0 percent interest? Would that be a great deal? Would people rush to do that? This is the equilibrium we solved over here already. So is that a great deal? Would that upset the equilibrium? Would anyone bother to take the Chinese deal if they lent at 0 percent interest, they were offering to do that? What?
Professor John Geanakoplos: They wouldn’t take it? We’re back here. What’s the nominal interest rate in this economy?
Student: 200 percent.
Professor John Geanakoplos: 200 percent interest, so if you want to borrow in this economy from another American you have to give the guy 200 percent interest. Here the Chinese are offering to lend you at 0 percent interest. So, yes, everyone would rush to take the thing and that would have a big effect on what the equilibrium was if the Chinese were willing to lend money at such a low rate of interest.
Let’s try another question. Suppose you invented a technology, new technology, new technology turns 1 unit, 1 apple today, into 2 apples tomorrow. Is this something people would rush to do or not? Suppose some inventor figured out how to do that, would he rush to do it? Could it be used to help the economy? So let’s put it this way. Could this new technology be used to make a Pareto improvement, everybody better off? Yep?
Student: That’s no, because an apple tomorrow is worth less than half of an apple today. It’s worth a third of an apple today, so no one would want to do that.
Professor John Geanakoplos: So that’s exactly the right answer. Actually you’re answering two questions. I asked two questions. One is could it be used to help the economy, make everybody better off? If a social planner was in charge of things and the Chinese invented this new technology, or some American in Alaska invented this new technology should the government use the technology and could it use the technology to make everybody better off, and the answer to that is no.
And then the answer to a second question is–suppose the guy in Alaska discovered it himself. He couldn’t care less about the Pareto improvement and helping other guys or the American planners or anything, he just wanted to make a profit for himself. Would he make a profit? The answer is no, because the real prices, Fisher would say, are 1 and a third and no matter how you look at it the interest rate is 200–he’s losing money, because he’s giving something up that’s worth 1 and he’s getting something that’s only worth 2 thirds. So he’d be losing money to do it. He’d lose money.
So we could prove that even. So the answer is no. That’s the first question, and nobody would do it anyway. And fortunately nobody would choose to do it–choose to use it–because it loses money. So those are two separate questions. Could it be used and would any individual choose to do it? Would it be good for the society and would any individual choose to do it? The answers happen to agree here. So why can’t it be used as a Pareto improvement? What’s the proof of this that it can’t be? The answer’s no. What’s the proof?
Well, the proof is that if it did, if in the end it led to an allocation X-hatA1, let’s call it X-tilde A1, X-tildeA2, and XtildeB1, XtildeB2 that made everyone better off. Then, well, we give their old proof. Then what? It means that P1 XtildeA1 + P2XtildeA2 is bigger than what? P1 E-hatA1–(all right, that’s what you have in the Fisher economy) + P2 E-hatA2 and similarly P1 X-tildeB1 + P2 X-tildeB2 is bigger than P1 E-hatB1 + P2 E-hatB2.
So why is that? Because in this Fisher economy, the general equilibrium–if this allocation really made A better off than what he’s gotten, than 4 third and 2, he would have chosen it. And B, she would have chosen her thing if it was better than 2 thirds and 2. So clearly they must have been too expensive for those two to choose because they were rationally choosing the right thing given what they could afford. So then you just add the stuff up. You add and you find that total consumption value is bigger than total endowment value.
Chapter 4. Effects of Technology in Fisher Economy [00:39:25]
That’s in the Fisher economy, but we’ve changed the Fisher economy because now we’ve added this technology which took away some of the first good and made it into the second good, but that technology just lost money, which is bigger than total value in new technology economy, right?
And so that’s a contradiction because the consumption of this, however the new technology got used in the end the total consumption of the people had to be the total of what there was and what was produced in the economy. The value after the new technology is introduced in that new economy has only gone down compared to the Fisher economy, and the Fisher economy value of endowments must have been less than this brilliant new allocation, and that’s a contradiction because this new allocation has to add up to the stuff that’s there in the new technology economy.
So that’s how we know that no new technology could possibly make everybody better off, and we know trivially it makes everyone better off if and only if it makes a profit. So if and only if it makes a profit can it be used to make everybody better off, and amazingly, in a free market economy, people are going to use it if and only if it makes a profit. So they’re going to use it if and only if it’s a good thing for the economy. So that’s the basic laissez-faire argument–that there are new discoveries all the time. Every other day somebody’s thinking of something new. Are we going to use it?
Should we use it? Is it something we need to read about in the papers and use? Well, there are a whole bunch of people, the discoverers themselves they’re going to talk to their business friends, and they’re going to say, “Do you want to lend me the money to get this thing going,” and all of them are going to do this profit calculation. If they decide it loses money they’re not going to do it, and thank god for that because it couldn’t have helped everybody if they did use it. So that’s the main lesson of laissez-faire.
So let me just put this in perspective a little bit. In the old Russian economy of the 1930s and ’40s there was no profit system, so the central planner had to figure out, should a new invention be used or not. So every time there’s a new invention a committee had to get together, of central planners and decide whether to use it or not. And there’s a famous guy named Kantorovich who was in charge of a lot of that. He won the Nobel Prize in economics. He shared it with a Yale economist named Koopmans and so Kantorovich told this very amusing story.
He said that there were two central planning bureaus. One was in charge of allocations and one was in charge of prices. One had to set the prices. The other had to set the allocations. And of course the whole message here is that you have to combine these. You don’t know whether it’s worthwhile to change the allocation until you know whether the new technology’s going to make a profit or not, and here they had the two things separated. They were telling people what to do before knowing whether they made a profit or not because they didn’t have prices because there weren’t free markets.
So the bottom line of the Fisher story is that you take this complicated financial economy, you reduce it to something very simple that you learned how to do in your freshman year or your sophomore year, solve that, and you go back to this and you can understand a lot about this economy. That’s something that most people didn’t realize at the time and still don’t realize now.
So you ask a typical person if there’s inflation, that means the dividends next year is going to be higher, is that going to raise the value of the stock today? Just like he said, “Yes of course because it makes the price of the dividends higher tomorrow.” Fisher would say no, it doesn’t change anything real in the economy. If there’s more inflation there will be a higher nominal interest rate, so discounted by the higher interest rate payoffs of the stock will give you the same stock price as before. So we’re going to do a thousand examples of this, but are there any questions about this? Yes?
Student: Can you just review your arguments at the end? I’m just having a very hard time reading.
Professor John Geanakoplos: Yeah, sorry. I don’t know if this is in the way, by the way. So this is the argument we gave a few classes ago. I forgot when. We said, how do you know that a final allocation that emerges as a competitive equilibrium is Pareto efficient? And the argument was if you can do better–that means, make everybody better off–then each person, if you look at the value of what they’re getting under the new regime it must be more than the value of their endowments otherwise they would have chosen the new regime and nobody chose it. That means everybody would have had to pay more for this new regime allocation than the value of their endowments. So this is more than that for person A, and person B’s consumption is more than the value of this endowments, his extended endowments in the Fisher thing under this new regime, than the value of his endowments. You’re following that?
Professor John Geanakoplos: Then the next step was to add all this up. Now notice, however the new technology affects the world, obviously people can only eat what’s being produced. Everything that’s being produced is part of somebody’s endowment. So if the new technology, if Mr. A invents the new technology, he gives up some of his good at time 1 to get more of the good at time 2, so his endowment has changed–but he’s got a new endowment, but it’s still his endowment. So whatever the new allocation is it has to add up to the new endowment.
Now, I haven’t even bothered to write down the new endowment, but I know the value of that new endowment. Whatever it is, it’s going to be less than the value of the old endowment, because the new technology loses money. So the contradiction is the value of the new endowment after the technology is used, at the old equilibrium prices, is lower than the value of the old endowment at the old equilibrium prices. But that, since it’s true for every person in the aggregate, that’s less than the value of this new regime consumption. And that’s a contradiction because the new regime consumption, that’s all this stuff, has to equal exactly the total endowments in the economy to begin with, and that’s the contradiction. So you can’t make everybody better off.
That simple argument, which as I said, my advisor Ken Arrow, another guy at Yale named Gerard Debreu–both of them were working at the Cowles Foundation which is part of Yale–that proof that they gave is the simplest and most important argument in all of economics. So we get as a conclusion that, putting it another way, that owners of firms should maximize the value of their firms, the stock market value of their firms, and thank God they do because if they find some new way of producing that’s going to lose money it’s going to make the stock market value go down. Remember the stock market value is just the same calculation, the value of all the output they’re producing. If they find some way of losing money and they try to use it it’ll make their stock market value go down. That’s why they’re not going to do it, and thank God for that because it’d be a bad thing for society if they did do it. Yes?
Student: Well, it seems to me this proof is logically flawed because you’re assuming that after the inception of a technology the prices are left unchanged, but that might not be true. Shouldn’t you have some argument for the prices not changing after the inception of the technology?
Professor John Geanakoplos: This is a very bold question, telling me that it’s a flawed proof. I want to commend you for your courage. As it happens, however, you’ve asked the same question that somebody asked a class or two–which is a very good question. So the answer is no, I shouldn’t have changed the prices and that’s exactly the point of the proof. So, yes it’s true that after the new technology is introduced the prices changes, everything changes, but we don’t have to worry about all that complication. After all the changes there’s going to be some final allocation of goods that supposedly makes everybody better off. So I can ask the hypothetical question. Would this new allocation to A at the old prices be something he could have afforded, and the answer must be no…
Student: All right, I’ve got it.
Professor John Geanakoplos: Well, let me just finish. You see the answer to your question, but I’m going to say it out because it’s a very important question. The proof is clever precisely because of what you’re asking. You have to do something that you wouldn’t have thought of. You have this new economy, and new allocation, and new prices, but the proof says let’s do the hypothetical thing of looking at the new allocation at the old prices.
At the old prices A couldn’t have afforded this new allocation because if he could have, he would have bought it because it makes him better off. So at the old prices A couldn’t have afforded this new regime allocation. Similarly B, at the old prices, couldn’t afford this new regime allocation. So at the old prices everybody would have to be spending more on the new regime allocation than the value of their endowment.
That means at the old prices, the total in the whole society–by adding it up–of the expenditures on the new regime consumptions must be bigger than the total value of the old endowments.
Now that was the contradiction why at the old endowments without production you couldn’t make everybody better off. We’d already have a contradiction. Now we add one more step. We’ve got this new technology that changed the old endowments. It changed the old endowments, but however it changed it we don’t have to keep track of how it did it. It makes the value of the total endowments even less than it was before, so we actually get a worse contradiction than before. So it was a good question, so I thank you for the question. Any other questions? Yes?
Student: Can you raise the board a little bit?
Professor John Geanakoplos: Yes, I can raise which board, not this one?
Student: Yes, that one.
Professor John Geanakoplos: Yeah. Well, sorry.
Professor John Geanakoplos: So the bottom line here is that–let me just summarize. We’ve spent four classes on reviewing standard intermediate micro and macro. People never talk about that stuff when they do financial–finance courses, in typical courses. However, Irving Fisher, the inventor of half of finance, that’s how he began. And it’s going to turn out now, especially in light of this last crisis, that the best way to understand what’s going on is to go back to the original underlying economy.
So Fisher said you can always take–we haven’t introduced risk, by the way. When that happens things are going to get more complicated. Fisher couldn’t deal with risk. So without risk, where everybody’s anticipating the dividends in the future, that means that you can always reduce a financial economy up there to a general equilibrium, which you’ve been taught before you got to this course, most of you, how to solve.
And now that solution to that problem with marginal utility and Pareto efficiency that tells us an enormous amount about how the stock market and everything works. It tells us that the value of every stock is just the discounted real dividends, discounted at the real rate of interest, or the discounted nominal payoffs, cash flows, discounted at the nominal rate of interest.
And it tells us that the real rate of interest is the nominal rate divided by the rate of inflation. And it tells us that it’s a good thing all these owners of companies are maximizing profits or share value, which is the same thing, and that’s helping society. So that’s the lesson. A lot of that stuff is going to change a little bit, but that’s the basic idea.
Chapter 5. The Impatience Theory of Interest [00:51:31]
So finally let’s get to the point. For 2,000 years the public was confused about interest. They said–Aristotle, one of the greatest geniuses of all times, he thought interest was an unnatural act. It was horrible even though, of course, lots of people in Greece were charging interest. Delos, the Delphic oracle was charging interest, would lend money at interest, and Aristotle and everybody was talking about the Delphic oracle all the time. They weren’t even paying attention. The Delphic oracle was charging interest and they were saying it’s totally unnatural.
So three religions all thought interest was a terrible thing. They all thought the just price was–the nominal rate of interest should be 0, but what Fisher says is the nominal rate of interest is irrelevant. Nobody cares about the nominal rate of interest. They look at apples today and apples next year. The money and stuff just gets in the way. It’s the real rate of interest that you care about, and the real rate of interest doesn’t have to be positive. It could be negative like it is today.
The real rate of interest, what are the determinants usually of the real rate of interest if the Federal Reserve isn’t mucking around with things, the real rate of interest is obtained by solving for P1 and P2 in this general equilibrium model. So what would change the real rate of interest? All you have are the utilities and the endowments. So here’s the economy. What would change the real rate of interest? So the first thing Fisher says is impatience. So in fact one of his most famous articles is called an Impatience Theory of Interest, so let’s call it that, Impatience Theory of Interest.
So Fisher said that in his view people are impatient. Why? That means an apple today they thought was more valuable that an apple next year. Why? Because of the poor imagination, it was easy to think about eating the apple today. You can just hold it in your hand and it’s so close, but to think about eating it in a year requires some imagination. They had poor imagination, and secondly, the second main reason is mortality. They might die between today and next year.
So those are the two main reasons. He gives a bunch of others, which I’m going to mention shortly, but these are the two most interesting ones, poverty of imagination and the fact that you just might die in between. So what does it mean? An apple next year is not a sure thing. There is the Impatience Theory of Interest. So he said that’s why it makes sense to have this guy A as impatient because he values the apple today more than a value tomorrow. He’s got this discount rate, a half here. B’s not impatient because the discount factor is one.
So he put a discount factor–actually Fisher didn’t quite have a discount factor, he had a more general thing, so Samuelson was the one who introduced the discount factor. It doesn’t matter, but anyway so a discount factor to capture Fisher’s idea that the good next year, the same apple next year is not worth as much to A as an apple this year. So suppose I change a half to a third? What will happen to the real rate of interest? So that makes people more impatient. Why does it make them more impatient, because now they care even less about the good next year. So when did this happen? In the Reagan years, the now generation, everybody talked about the now generation. People are getting more impatient. So what happens to the real rate of interest when people get more impatient? Does it go up or down?
Student: It goes up.
Professor John Geanakoplos: So why does it go up? That’s correct.
Student: Because there needs to be more of an incentive to save.
Professor John Geanakoplos: Right, but now Fisher would say that a little bit more–he would say it a little more formally, but that’s exactly right. In order to get anybody to save, because they want the stuff now, you’re going to have to give them a higher real rate of interest. That’s exactly right. So how could you say it in this economy? [next slower]
Remember in this economy, this Cobb-Douglas economy, you could prove it formally. You know that if P2 (let’s say) = 1 and we’re solving for P1 and here’s the supply, this is X1, and here’s demand. So remember XA1 is going to be something like P1 EA1 + P2 EA2 times 1 over 1 + delta where delta–what’s called delta, the discount. Let’s call this delta, so the discount.
So to get these to add up to 1 I take 1 + delta. So the weight on this thing is 1 over 1 + delta times this divided by P1. So if P2 is 1 then this is just equal to 1 over 1 + delta times (EA1 + 1 over P1 times EA2). So clearly the demand goes down as P1 goes–as P2–this is P1, so P2 = 1, so if I divide by P1, P1 over P1 goes away. Then I have P2 over P1, and if P2 is 1 that’s just 1 over P1. So obviously as P1 goes up your demand goes down. That’s just what you’d expect. So P1 goes down the demand goes up, or P1 goes up the demand goes down.
So anyway, if you add up Cobb-Douglas people it always is like that. The demand for any good goes up as the price goes down, if its own price goes down. So if you change delta, if you make delta smaller, that’s going to raise demand for A1 at the old prices. Why? At old equilibrium prices, the same trick as before, at old equilibrium prices what’s going to happen? Delta goes down like we just said, implies XA1 goes up. So the guy’s demanding more now, but if he’s demanding more at the old equilibrium prices–so at the old equilibrium prices he’s demanding more so the only way to clear the market is to raise P1. Implies P1 must go up to clear the market. So this is a formal proof of what he just said.
So the common sense maybe is enough for you. If you care less about the future to get anybody to save you’re going to have to raise the interest rate. To say it formally if we solve for equilibrium with a lower delta at the old equilibrium prices, this guy at the old prices, A would now shift and try to demand more of good 1. But if he demanded more of good 1 that would mean too much demand for good 1, and the only way to clear the price of good 1 is to raise the price P1. But if you raise P1 holding P2 fixed that’s just P1 over P2, so the interest rate, so the interest rate has to go up. So that’s your argument made formal.
So that’s his Impatience Theory. That’s the main determinant of interest according to Fisher. What’s the second one? He says suppose people are more optimistic about Ei2? Everybody thinks the world’s going to be much better next year. We’re going to have more endowments. What do you think is going to happen to the interest rate, the real interest rate, somebody else?
Student: It’ll decrease.
Professor John Geanakoplos: It’ll what?
Professor John Geanakoplos: Decrease, why?
Student: Because you’re expecting things to be better <> signifies people will save less.
Professor John Geanakoplos: To save less or to save more? So let’s think of good X1. If people thought they were going to be richer at the old prices what would they do today for X1 demand more or less today?
Student: The rate would go up, right?
Professor John Geanakoplos: Yeah, the right answer is up. He said down, but let’s just figure out why.
Student: They demand more <>.
Professor John Geanakoplos: So the reason I gave the formal argument is because you can get confused here. So let’s just do the intuitive one. So you had the idea back there of the intuitive one, you just got it backward, but you were on the right track. The point is there’s going to be so much stuff around for people to eat tomorrow, you’ve got to get them to want to eat all that extra stuff tomorrow. So you have to give them an incentive to want to eat all that extra stuff tomorrow, so you have to raise the interest rate, not lower it. So you had the right idea, the wrong conclusion.
Now, how can you actually give a formal proof of that so you know you’re not confused? Again, like his question, at the old prices what’s going to happen to the demand for X1? At the old prices, since you’re going to be so rich in the future, you think you’re just incredibly rich now, so of course you’re going to consume more today. So there’s going to be more demand today and the endowment today hasn’t changed. So there’s going to be more demand today with the same endowment today, so therefore in order to clear the market today you’re going to have to raise P1 relative to P2 so the interest rate’s got to go up. So is that clear?
It’s a little surprising, so let me say that again. If you increase the endowments tomorrow the supply today of goods hasn’t changed, but people are richer tomorrow. So clearly they’re going to consume this fraction of their wealth. Their wealth is up. You tell anybody, “You’re going to be rich next year. You’re going to be worth a fortune,” the normal person, Cobb-Douglas person, is going to consume more stuff today anticipating that he’s going to be so rich tomorrow. He’s going to borrow against tomorrow’s wealth. And so therefore, in order to clear today’s market where the supply hasn’t changed, with all these people trying eat more today you have to raise today’s price relative to tomorrow. That’s, raise the real interest rate.
So what’s a third example? This is Fisher’s most famous one. Suppose you transfer money, transfer wealth, from poor to rich. What would happen? We have to make an extra assumption here. Fisher felt that the people who were rich were rich because they were patient. They could charge interest and get lots of money. So if you change wealth you take away some money from the poor. That’s what’s happened in the American economy over the last 15 or 20 years. The rich have gotten richer and the poor are pretty much back where they were before. So suppose the rich get rich at the expense of the poor? What’s that going to do to the real rate of interest? I’ll–hang on a second. Yep?
Student: That would make it lower.
Professor John Geanakoplos: That’s going to lower it. Why is that?
Professor John Geanakoplos: So there’s an intuitive way of saying it which is his which is that the rich, because they’re patient, are probably the lenders. Now they’re even more willing to lend and so the interest rate has to go down to get these other people to borrow.
A formal way of saying it is that if you transfer money from the rich [correction: poor] to the poor [correction: rich] that means the poor guys–the rich guys always consume a higher proportion in the future because they’re more patient. So a more patient guy will consume more in the future. So if you take away wealth from an impatient guy and give it to a patient guy you’re going to increase the–the economy’s going to be more in the hands of the patient people, and so the patient people–the mix is going to change. People on average are more patient than they were before so on average in the economy they’re going to consume less than they were of today’s good and so the shift is going to be in this direction, right?
Because you’ve made people, a lot of them impatient, a lot of patient, you’ve increased the patient ones and decreased the impatient ones, so in balance you’re going to decrease demand today because it was the impatient ones who wanted to eat today and the other guys were willing to wait.
Now the guys who aren’t willing to wait these guys don’t have any money. They’re the ones doing all the consuming today and now they can’t afford to do much consuming, so you’re going to reduce consumption today. So to get the market to clear again you have to lower the interest rate this time. So those are three famous conclusions of Fisher, more impatient people, higher interest rate, more optimistic about the future, higher interest rate, transfers from the poor to the rich lower interest rate. So what happens to the stock market in this case? Suppose people are more impatient. Does the stock market go up or down?
Professor John Geanakoplos: Down, because the stock market price is just this, the real interest rate times the dividends. So I haven’t told you the dividends changed, so if the dividends are the same and the real interest rate has gone up the stock market has gone down.
Suppose people are more optimistic about the future, so not about the stocks producing more, but about whether there’s more stuff in the world? Their own endowments will be bigger. The stock market is going to go down. That ones a little subtler because they could be optimistic about the stocks producing more, so that’s ambiguous. So let’s do the third. Suppose you transfer wealth from the poor to the rich, what’s going to happen to the stock market? It’s going to go up. So what happened in the last 20 years? The rich got richer, the poor got poorer, the interest rates got lower and lower and the stock market got higher and higher just as Fisher would have said.
Chapter 6. Conclusion [01:06:48]
So I want to now end with just Fisher and Shakespeare, so I’m going to go over just a couple minutes. Maybe I’ll have to start with Shakespeare. So Fisher’s theory of interest, as I said, was making sense of thousands of years of confusion, so the idea is that interest is nothing other–you shouldn’t think of nominal interest. People look through all that. They look at the real rate of interest and the real rate of interest is just the ratio of two prices just like everything else in equilibrium, so therefore there is no such thing as–it’s an important price like anything else, but maybe I forgot to say it, there’s no such thing as a just price.
The price, in fact, that equilibrium finds is the best price because that’s the price that’s going to lead new firms and inventors to use technologies that help the economy as opposed to hurting the economy and wasting resources. So the price that the market finds is the just price and the real rate of interest is the right real rate of interest provided that people are rational and see through this veil.
So, why is it that the real rate of interest is typically positive? Well, it’s because, as I said, people are impatient and these different reasons. Now Fisher said one other reason that screws up the real rate of interest is people sometimes get confused by inflation.
So this is an aside. He said that all contracts should be inflation indexed, and he forced his Yale secretary and his secretaries at his company to change their contracts–I guess his Yale secretary is probably wrong, the secretaries at his business, Remington, he forced them to accept deals where their wage was indexed to inflation. And of course the Great Depression happened and all of the prices collapsed, and so all his secretaries got less money out of the deal so he wasn’t too popular with them either.
He says impatience is a fundamental attribute of human nature. As long as people like things today rather than tomorrow there’s going to be interest. So interest is, as it were, impatience crystallized into a market rate, and the reasons for impatience are this foresight, lack of foresight, possibility of dying and then he talks about self control and stuff like that, the greater the foresight, etcetera.
Now he has this racist view of the world, which I think is worth mentioning. So he compares the Scotch and the Irish, so the Scotch are patient, the Irish are totally impatient, no self-control and it gets worse and worse. I can’t show you all of this. So Holland, Scotland, England, France these are all the places his family was probably from. They’re incredibly patient. They’re wonderful. They’ve got low rates of interest, incredibly thrifty people. Then you look at all these other dreadful people, Chinese, Indians, Blacks, Java Southerners, American Indians and then Greeks and Italians he mentions later, hopeless, high rates of interest, incredibly impatient.
So anyway, the patient accumulate wealth and by waiting and lending they make production possible, because the people with all the good ideas where are they going to get the money to produce? They’re going to get it out of the patient people who are willing to wait. If you can wait should I talk for five more minutes or do you need to go? I was going to do my–maybe I should let you go.
Anyway, so what I was going to say last, I won’t say it, is that Shakespeare anticipated all of Fisher’s Impatience Theory of Interest and went a step further. He said, “Well, that’s great but you should take into account that people won’t keep their promises, and if they don’t keep their promises you need collateral, and if you need collateral that’s going to change a lot of stuff,” and Shakespeare already had a lot of that figured out, and most of this course is going to be about, believe it or not, what Shakespeare had to say about the rate of interest and collateral. Okay, next time.
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