ECON 251: Financial Theory
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ECON 251 - Lecture 13 - Demography and Asset Pricing: Will the Stock Market Decline when the Baby Boomers Retire?
Chapter 1. Stationarity and Equilibrium in the Overlapping Generations Model [00:00:00]
Professor John Geanakoplos: All right, so I’m going to finish talking about overlapping generations today. I’m going to defer my plan to save Social Security till a couple of lectures from now, and I’m going to start to introduce uncertainty in this class.
The exam will only cover the stuff up to basically the last problem set. There are a few things about overlapping generations I might say today, which will clarify what you already know, but you don’t have to know anything beyond the problem you turned in on Thursday–you’re turning in today. So you I think are better informed now than you were yesterday. I think it’s a big help to be doing the problems, so for those of you watching on screen at home, do the problems.
So remember we had a situation where there were generations, each of whom was rich when young and poor when old, that looked like that. And it went on forever. That was the only difference from what we’ve had before. People’s lives were shorter than the lifetime of the economy, and to go to the extreme, the economy goes on forever, even though people only live two periods.
And then there was land, 1, 1, 1, 1. There’s land, and then there’s generation–because I forgot the generation–here’s generation 0, generation 1, generation 2, generation 3… and time. This is time. And here’s time 1, time 2, time 3, time 4, time 5, etc. Now this model, I’ve been told, is the hardest thing you do the entire class. Things are going to get much more complicated, but since they build up slowly, I think this model’s the hardest thing you have to do the entire class.
I realize it’s a little bit confusing, but in the end, I don’t think it’s that complicated. And it reveals all the subtlety of Fisher’s thinking, not that Fisher understood about overlapping generations, but his ideas, you can see here.
So what are his ideas? You’ve got a complicated model in which you’ve got people trading apples when they’re young, against land. So the objects in the economy are land, which you’re supposed to think of as the stock market, okay, so we’ve got land and we’ve got apples.
Now what’s interesting is, the land is going to constantly change hands. So the young here are going to buy the land, but when they get old, they’re going to sell the land. So the land is constantly changing hands. It’s not that one person buys it at the beginning and holds it forever.
So Fisher’s formula, remember, was that the price of any asset was going to be equal to the discounted dividends. And the logic seemed to be at the time that the price, you know, if the person was going to buy the asset, that they could take the money now or they could wait and get all the dividends in the future, and they would compare what they would get out of the dividends in the future to what they have to give up today.
But you see, in this example, it’s a little more complicated. Nobody’s going to hold the land and get all the dividends. Each person who buys the land knows that he’s going to sell it long before getting all the dividends from that land. And yet the price of the land is still given by this formula, so we never use the fact, in Fisher’s argument, that the same person had to hold the asset till the end.
And so now we have a case, an extreme case, where people only hold the asset for one period. It’s always turning over, turning over, turning over, but still the price of the asset is equal to the discounted dividends. So the secret to solving this problem, according to Fisher is, forget about the asset. That makes it too complicated.
Fisher’s argument again is, take financial equilibrium–maybe I should write it on a different board–Fisher’s argument is, financial equilibrium, equilibrium, okay. Get rid of assets. Put dividends into endowments, and transform it into general equilibrium, the very thing you saw within the first couple days of class, okay, and then go backwards and figure out what the financial equilibrium is. So although it seemed very easy before, it’s a little subtler this time, but it’s the same principle.
Whose endowment should you put the dividends into? The land is changing hands all the time. There’s a new guy owning it, a new guy owning it every period, so whose endowments do we increase when we put the dividends of the land into someone’s endowment? You put it into the endowment of the guy who owned it first. So put dividends into endowments of original owners, okay?
So this old person who has the right to the first apple, his own endowment of an apple, and also the right to the first dividend, but he actually has a right to all the dividends. So his endowment, the endowment of Mr. 0 is going to be 2, 1, 1, 1, etc. Because he has his original apple, plus he’s got–he has his endowment of an apple, because everyone when they’re old has an apple. He’s also, because he’s owned the land before, gotten the right to this first dividend of the apple, that’s 2, and because he’s the original owner of land, he owns all the future dividend streams of the land.
So we can forget about land after we’ve stuck the dividends into the right person’s endowment, and now we just trade as if it’s general equilibrium. We just look for supply and demand and the present value prices, which are p1, p2, p3, etc., okay? And because it’s symmetric, we know that this is going to equal 1–p–we can always make one of them be–one price can be 1, so the present value prices are going to look like this.
The reason I can make this simplification is by symmetry. Every generation is the same, so it’s obvious that I’ll be able to keep repeating the same argument over and over again. Okay, and so what do I have–so remember, the land has disappeared here.
Even though this generation is buying the land and selling the land, in Fisher’s story, you don’t have to pay attention to that. This guy bought the land, and then he sold the land, so it was a fair trade. So he started without it and ended without it. The fact that in between he used the land to save, you can already do that in a general equilibrium. You don’t need to think about the land, because the only reason this guy is buying the land is to use it for a means of saving, which is already in Fisher’s equation.
So to solve for equilibrium is then quite simple. We just, in every generation, we look at–remember, I forgot to write the utility functions, so we have U of consumption when young and old is 1 half log young [consumption] + 1 half log old [consumption]. Okay, and so therefore, in equilibrium, we have to have that 1 half times the income of every generation–every generation’s going to have 3 when young + 1 when old.
Now from the point of view of any generation t, the prices are p, say, p2 when young and p3 when old. Now remember, any generation cares only about relative prices. So I can call them p2 or p3, or I can call them 1 and p, because the ratio’s going to be p. So, much simpler to think of the prices as 1 and p.
So the old guy, if I’m doing any t, like this period 4, the old guy here, you know, that’s p4 and p3. He cares about p3 and p4. But he only cares about the ratio of p3 and p4, which is p. So from his point of view, he might as well have thought of them as 1 and p. So his income is 3 + 1p, and he’s going to spend half his income on that good, which for him is a relative price of p. The young guy is going to spend half his income, which is also 3 + 1p. This is the relative price between t - 1 and t. This is the relative price between t and t + 1. But he’s now young, so his price when young is 1, and that has to = 3 + 1 + 1 =– so I wrote it as 1 + 3 + 1 = 5, okay? And we solved that, and we found that p = .55. So are there any questions about how we did that? That’s where we ended the last lecture. Is that clear to everyone? Yes.
Professor John Geanakoplos: Okay. If I didn’t have symmetry, I would have to say this is–I would have to write this–in general, this would be 3pt-1 + 1pt over pt, and this guy would be 3pt + 1 pt+1, divided by pt. Right? That would be, in general, if I had prices, the present value prices, this guy’s income would be given by that, because he’s the old at time t, and the young would be given by that, okay? And I’d have to do this, a new equation, for every t. Who asked? You asked the question, so does that make sense?
Professor John Geanakoplos: Okay, but I have to do that for every single t, so I have an infinite number of prices and an infinite number of equations. But then you say to yourself, “Well, hmm, if I take the ratio of pt over pt-1, to = p, then the ratio of these two prices is the same as this price,” and so I’ll have one equation and one unknown variable, and that same equation will get repeated every other generation.
Because when I go to generation s, which is another generation, all that matters is the ratio between pt-1 and pt, and ptand pt+1. So if I assume those ratios are always the same, I’m going to be repeating the same equation over and over again. So if I can solve it once, I can solve it an infinite number of times. It’s basically the same equation written infinitely often, and so as long as I stick in–keep these ratios constant, I’m just doing it over and over again, so I only have to do it once.
So it doesn’t have to work out that it’s–it’s just because every generation has the same endowments and every generation has the same utility. That’s why it’s such a simple thing to solve. Yeah?
Student: With the regular p, before you erased it, is that a weight thrust that becomes the old <> because we assume young age <>?
Professor John Geanakoplos: The p, this p, is just the ratio between–this is how I got p. pt–okay, p is pt over pt-1, which also = pt+1 over pt. The ratio of all the prices is always p. The ratio between time t and time t + 1, this good is more valuable than that good. This good is only worth p. p is .55. It’s only worth 55 percent of this good.
And then this good here is only worth 55 percent of that good. So the symmetry assumption, the symmetry of the problem suggests that I could guess that the prices are–the present value prices are going to decline by the same percentage every period. Okay, any other questions? We’ll keep going with questions. Yes?
Student: Could you explain the e0 = 2, 1, 1, 1?
Professor John Geanakoplos: Yeah, what is the endowment of the original old guy? He’s the guy who owns the land. So what does he have? He has one apple, his endowment when he’s old, like everybody else who’s old.
He has an apple, his own tree in his back yard gives him an apple when he’s old, or when he’s an old man or an old woman, he can barely scrape together one apple. Now he also has the land. The land is his for all time, so the land is paying an apple every year forever. So his own apple, at time 1, plus the land’s apples is this 2, and the land produces an apple every time in the future. So that’s his endowment. I stuck the dividends of the land into his endowment.
Student: <> like he lives forever, or <>?
Professor John Geanakoplos: Right, I’m not assuming he lives forever. He doesn’t. In fact, he knows he’s going to die in period 1. He wants to do all his consuming in period 1, but he hasn’t–that’s another subtle thing about this. You can own endowments that occur after you’re dead. If you own a tree, the tree, the land is going to be producing stuff long after you’re dead, so when you sell off the land, you’re selling off those future dividends.
Of course, you’re not going to hold the land. If you don’t care about your children, you’re not going to hold the land beyond the time you live, because you want to just consume as much as you can. So you’re going to sell off the land and consume, but you’re selling off all these future dividends, okay?
So another thing this model gives us is a new argument for the real interest rate, real r, being greater than 0. So why is it? Because from this formula–gosh, I was so clever. Where did I–oh, here. From the price of the asset, this discounted dividends, that’s going to be 1 over (1 + r), + 1 over (1 + r) squared, + 1 over (1 + r) cubed, + …, by the way, that’s = to p + p squared + p cubed, you notice, okay?
So the discounted value of the dividends is the price of land. So the price of land, of course, has to be finite, because if the old guy owned a thing worth an infinite amount of money, he’d sell it and he’d buy everything in the world. So if you’re going to have equilibrium, there’s no plot of land that sells for 10, for 100 trillion dollars. The price of every plot of land is some finite amount.
So if the land is producing, I think it’s sort of realistic. Land is probably going to produce forever. As long as the world lasts, if you’ve got land, you’ll get something out of it. Someone will always want the land, and so if it’s going to have a finite price, the whole future dividends have to be discounted to a finite number.
If you put 0 in here for r, you get infinity. So this presence of land is forcing the discount rate above 0. So it’s another argument that Fisher never would have thought of, because he never thought about infinities or anything like that–another argument for the real interest rate being positive. Okay, any other questions about this?
So we reduce it to one equation, and I’ll put back the p here in just one unknown. Okay, = 5. And we saw that we get p = .55. And it’s a quadratic equation, so you can all solve quadratic equations. Okay, what else do I want to say?
Chapter 2. Evaluating Tobin’s Thoughts on Social Security [00:16:38]
So now you could analyze all sorts of things. You could do the same comparative statics Fisher does. Suppose we change the utility function and made this 1 third instead of 1 half. That would correspond to doing what? Making people more? More what? Impatient, okay. And what would you think would happen to the interest rate?
Student: Go up.
Professor John Geanakoplos: Go up, okay. So in the Excel file, we’re going to do just that. We’re going to see it’s going to go up. Instead, we could do the Social Security thing, and we could have every young person give something to the previous old, which means everybody’s endowment goes from say 3 to 2 and 1 to 2, so they get more in the future and less early on. So again, you could re-solve for equilibrium and Fisher’s saying, if you’ve got more endowment in the future and less in the present, you’re going to have a higher real interest rate. So you get a higher real interest rate.
You could make the land more productive and change this to a 2 everywhere, and see what happened. So in this Excel file, I do that. Maybe I’ll come back to that. So we’re going to do those experiments in a second, but now I want to do two more things before I end this.
The first is, I want to say, suppose we had growth in the economy. So what if you had growth? Where am I going to add growth? Over here. I shouldn’t have erased that. So let’s have growth. What happens? Well, Tobin, you know, Yale’s greatest economist after Irving Fisher, he said something which was 90 percent right but not 100 percent right.
He said, you know, I pointed out to him that I got these guys at Social Security. I worked with them. Anyway, these people at Social Security and the Social Security Administration calculated who were the biggest beneficiaries of Social Security, and it was Tobin’s generation, the same generation as my father.
And I said, “Look, it’s your fault that Social Security is such a mess. You benefited and I’m going to have to pay for it.” And just like you’re feeling you’re going to have to pay for it, so it’s even worse for you. But even my generation’s not getting a very good deal. Yours is getting a terrible deal, Tobin’s got a great deal. And so he said to me, it’s not his fault, it’s my fault, because I didn’t have enough children. So the question is, what happens if you–what does that mean?
So what Tobin meant is that if, instead of having 1 child for every adult–so every family has 2 children, in other words, instead of 1 child for every adult, so it goes 3, 1, 3, 1, 3, 1, we start doubling–the generations are growing through time. So since this is 30 years or something, we could think of the generations doubling. That wouldn’t be such a fast growth rate. And so Tobin might have meant, if you double the generations, you’re going to have 6 and 2 as the endowment. 2 guys at 3 and 1, which means 6 and 2.
Then you’re going to have 12 and 4. It’s going to be a gigantic growth, and so each generation, when it’s old, the young guys, if there’s 6 and 2 here now–I should use a different color. If I wrote 6 and 2 here, then each guy would have to–or 2 guys at 3 and 1, each (3, 1) guy would only have to give up half a unit, because there’d be 2 of them each giving half a unit to the old guy. And each one, when he gets old, getting a full unit.
So this guy would get a full unit. The next generation would only give up half a unit, and because there were 4 times as many people in the next generation, each of them would still give 1 half, making these 2 old guys get their full 1. And when they got old, they’d similarly get a full 1 each of them. So it would seem that if you had a faster growing population, then Social Security would be a much better deal, because the young would have to give up less to the old and still get fully paid back when they were old, and so Social Security wouldn’t have been a problem.
Ok, so why wasn’t it our fault, my fault–and your generation is probably not going to have any more children than my generation. Is that true, Tobin’s argument? Is it true that if we just had enough children, no one would have to lose out in Social Security? We could make the original generation better off and also every other generation after that better off? Yeah?
Student: <> even giving up half to get 1 wouldn’t <>.
Professor John Geanakoplos: Right, exactly. And so you can’t know that until you actually solve the model. So the point is, that if the generations keep growing like that, the interest rate is going to go up, and so it’s still going to be a loss, because although you’re going to be trading off - 1 half to get 1, as opposed to - 1 to get 1–so this looks like a much better scenario. Once you get to this scenario, there’s going to be a bigger interest rate.
So remember, the present value of that trade is - 1 + (1 over (1 + r)). That’s why it’s negative, because r is positive. Here it’s - 1 half + (1 over (1 + r)), but that’s a different r. That’s an r, I’ll put it at the top, that’s in a new economy, a growing economy, and that r is going to be a bigger r. So even though it’s - 1 half instead of 1, this thing’s going to be small enough, so you still end up losing money.
But you’d never know that unless you solved it. And in fact, it’s still going to depend on a little delicate assumption. So Tobin, by the way, never bothered to work this out. I claim that if you’re going to be doubling the population like this, there are going to be a lot more people working on the land, and so the dividends of the land are probably going to grow at the same rate. So I would add a 2 here. I’d make the dividends of the land also double, if the population doubled. So the land dividends are going to grow.
Now you can see why the interest rate has to go up, because the present value of the land still has to be a finite number, the present discounted value of the dividends. And if the dividends are growing, then the interest rate had better be growing even faster if you’re going to make that a finite number. So sure enough, when you solve it, you’re going to get a higher interest rate.
So how would you solve it? Let’s say the growth rate is g, so every generation is 1 + g times bigger the previous generation. So you could just change this equation very simply. What would you do? Well, for any generation, it’s the same equation, except that you’ve got 1 + g times as many old guys as you have–these are young guys. The number of young guys is 1 + g times the number of old guys.
So if it’s generation t, time t, this would be (1 + g) to the (t - 1), and this would get multiplied by (1 + g) to the t. And the dividend, 5, so everybody’s endowment is also going up, so it would be–actually I screwed up. It would be, the old guy’s 1 times (1 + g) to the (t-1) + the young guys times (1 + g) to the t, + the dividend times (1 + g) to the t, or t - 1, whichever. Okay, so that’s the new equation.
You can divide through by (1 + g) to the (t - 1). So this is period t’s market clearing equation, but you can see I haven’t–because the number of old guys, for each one of them that’s behaving like that, but there are now this many of them, and the young guys are behaving just as before, but there are this many of them, more of them.
And then the endowments of the old have been grown that far, you know, the old generation has this many endowments. The young has this much endowments when young, and then the land we also assumed grew. Okay, so now you can just divide through by (1 + g) to the (t - 1), and that goes away, and that just leaves that, and that goes away, and that goes away, and that goes away. So you’d have that equation.
Student: Can you do that again?
Professor John Geanakoplos: Oh no, yes. I was afraid someone was going to do that. So let’s just see where I got this equation from. So I took my original equation, which didn’t have any 1 + gs in it, and I just said, if I had growth in the economy, it’s no big deal to take that into account. I’m still going to have 1 equation, so I can still solve it.
It’s just that at every time t, how many old guys do I have? I have each old guy behaving the same way as before, except that now there’s been a growth of 1 + g. This is a generation before this one, so this generation is of size (1 + g) to the (t - 1). The young generation, which was 1 generation ahead, has grown by a factor of 1 + g. So there’s (1 + g) to the t of these guys. So I take their old demand, you know, the same for each guy, but different number of guys, and the young generation’s demand, and I have to set it = to the supply.
But that’s just the old guy’s endowment, so 1 for each old guy, and there are that many old guys, + the young guys’ endowment, same as before, and there are that many of them, + the land, which I’ve also assumed has been doubling, 1 + g is 2. So the land is also doubling. And I’m not sure whether I called this t or t - 1, but either one, it’s not going to make a big difference, okay?
Who asked the question? You follow that, right? I’ve got 1 + gs everywhere, so I can cancel a lot of them out. That’s all I did. So if I divide by (1 + g) to the (t - 1), I just get rid of the (1 + g) to the (t - 1)s. I change (1 + g) to the t divided by (1 + g) to the (t - 1). That’s just to the first power. I got rid of this one. I made this to the first power, and this to the first power, okay? So I can solve that perfectly easily and I’ll do that in Excel in a second. And then we can see what happens to everybody’s utility. So any questions about what I did here? Yes.
Professor John Geanakoplos: Yeah, it’s going to change the answer, but it’s going to qualitatively be the same.
Professor John Geanakoplos: Well, it depends whether when I do this I want to say this is 2, 4, 8, 16 or whether I want to call it 1, 2, 4, 8, 16. So I don’t know which is the right thing. Probably I should have done 1, 2, 4, 8. Probably the way I have it here which is t - 1. Okay, any other questions about this?
So the last thing to do is to just do it numerically, because it’s a pain to have to solve these, you know, these quadratic things. It’s only 1 equation, but it’s still a pain to solve, and I happen to have written a file that you can use to solve these all automatically. This doesn’t look too good. So the first one, the first case I did, suppose A, that’s the exponent for how patient you are. That’s half, that’s the first exponent, half, half. So I always can make the log utilities, the 2 coefficients, so it’s A log B. So I’m assuming utility here, U of young and old is A log young + (1 - A) log old, old consumption, okay?
So A I took to be 1 half. That’s just what we were figuring out. I took the endowments when young to be 3 and when old to be 1 and the land output to be 1 forever. Then the price is what I’m going to try and solve for, and so I solved the market clearing equation and cleared it and did Excel solver and got the consumption, which we figured out was 1.775 when young and when old was 3.225, which adds up, the two of them add up to 5 together, just as we thought.
And so I even calculated the utility of each of the people. I didn’t do log utility, actually, I wrote a monotonic transformation. I put Y to the A, Z to the (1 - A). It’s easier for me, you get positive numbers. That’s e to the U. So I might as well describe utility as the exponent of U. Take e to it, so it’s a positive number. When you do logs of small numbers, you get negatives, so that’s easier to look at. So that’s the utility.
Okay, now we can do Social Security. What happens with Social Security? The only effect of Social Security, remember, it’s the same model. You just take stuff away from the young and give it to the old. So nothing changes, except the young’s endowment goes down to 2 and the old’s endowment goes up to 2. And then the rest of the model’s exactly the same. And so, of course, it’s going to give you a different price. pA is different. So pA is smaller. That means today’s price of next year’s good is smaller. There must be more discounting.
So I did the interest rate, 1 over 1 + r = p, so the inverse of p is this thing. So the forward rate is 161 percent, and if you think of that as 30 years, that’s an annualized 3.2 percent interest rate. Before, it was only a 2 percent interest rate annualized, right? Remember we got the 1 + r to be 181 percent, 1.81, so r is 81 percent. So this Social Security doubled the interest rate from 81 percent to 161 percent. Huge increase in the interest rate, so Social Security has raised the interest rate.
And so you re-solve the whole equilibrium and it didn’t crush everybody, but everybody’s utility went down except for the very first generation. We know they won for free. But now I could do the same thing with growth. I could just solve that equation on the left, and so I plug in now the growth rate from generation to generation, 1 + growth, and here I’ve got 2.
So it doubles every period, just the example I did, and so I solve it and you see that of course from the generations growing and land is being more productive and there are more apples and stuff like that, there’s just more for everybody to consume. It’s actually made everybody better off. But now we see that Tobin and the interest rate is higher, way higher, 247 percent instead of 161 percent.
If I do now Social Security in the growing economy, every generation when young only has to give up half an apple and get 1 whole apple when old, so that raises the interest rate from 247 percent to 300 percent, annualized from 4 percent to 4.7 percent. So the rate of growth is faster now than the growth of the economy, right? This interest rate, 300 percent is faster than the rate of growth, which is 100 percent.
So again, everybody is going to lose. They’re going to go from 3.06–they’re still going to lose by Social Security, to go down to 3. But Tobin, I would say, is 90 percent right, although more children doesn’t solve the Social Security problem, because it’s going to increase the rate of interest. So you’re still going to have losers. Nonetheless, the loss is now much smaller with lots of children than it was before.
So Tobin, I would say, is 90 percent right. If only every generation would have more children, we still would have had a crisis in Social Security. We still would have had this every generation complaining it was getting a bad deal, but it would be a lot less bad than it was before. And then I did this variant, depending on what we said, whether the growth was–whether I had a g to the t or a g to the (t - 1) there. It doesn’t really change that much.
Okay, so that’s it. So you see, just by a very simple modification of the problem, you can resolve–it’s just understanding is, at least in my mind, so much clearer now about what role growth plays. Having more children doesn’t make the problem go away. It makes it better, but you can never make it go away, because the interest rate is always going to be higher than the rate of growth of the economy.
Chapter 3. Birth Rates and Stock Market Levels [00:35:07]
So let’s do one last experiment on this stuff. Another problem to resolve. I wrote a paper, the newspapers were saying the same sorts of things and I’ll tell you what they were saying. Oh no, I did a mistake. Sorry. Okay, so remember those pictures that we showed before of the stock market.
If you take out inflation, the stock market seems to go in these waves where prices go up and then they come down, then prices go up, then they come down and prices go up, then they come down. We’ve had 5 big waves of stock markets going up and coming down. We’re now in a period where they’ve been coming down. The stock market is lower now than it was 10 years ago, correcting for inflation. In fact, even not correcting for inflation. So why would you expect that to happen? Is there any logic to that?
So here’s an old picture of the stock market. I wrote this paper in 2005. Remember at the first class, we had a picture like that. You can see the stock market going up and down, up and down, up and down, up and down, five waves. It’s now come down further. This is like 2001 or ‘02, so it’s come down way further than that. So is there any explanation for why that would happen?
So here’s an explanation I gave in a paper, which a lot of people in the newspapers were giving. I said, the baby boomers. I said all these baby boomers like me, we’re getting old. We’re going to start selling our stocks and so the stock market’s going to go down. And so that’s what newspapers were saying too.
The difference is, that the economists criticized the newspapers and me and they say, “Oh you’re crazy,” they said. “The stock market, it can’t be that the stock market is going to go down, and everybody knows it’s going to go down, because if we knew it was going to go down in the future, it would have already gone down now.”
Okay, so you have to give an argument. So what I pointed out was that if you look at the history of live births, you get the same cycle. Look, it went up and down, up and down. I don’t know if you can read that. I can hardly read it myself. What years are these? Why is it so unreadable? Let’s just see if I blow it up a little bit. Now I can see it. This is 1910, ‘20, ‘30, ‘40, ‘50, ‘60, ‘70, ‘80, ‘90, 2000. You can see these were live births. So births, there a baby boom and then a plummet, then another huge baby boom. That’s the famous baby boom, my baby boom, and then a plummet. And then there was another baby boom. I forget what they call it, the X generation or whatever it is, some other baby boom that happened. Basically my baby boomers, me having children, another big baby boom and then birth rates are going to go down.
So I think that you are–the number of 18 year olds in the country is at its all time high, or is going to be in two years when my poor son tries to get into college. And so then it’s going to go down again. So this is a cyclical thing that keeps happening over and over again with baby booms and busts. And everybody knows about the baby boom after the war, my generation, the ’50s and ’60s when I was born.
Everybody knows about that, but they don’t realize that there’s been baby boom after baby boom after baby boom, and this is a cyclical phenomenon. And then the thing that they really don’t know is that it’s exactly the same as the stock market, that those 5 patterns in the stock market were the same patterns of the baby boom. Okay, so I’m going to end with that picture.
Okay, so a picture is worth a thousand words, but it’s not that compelling an argument. You need a little bit of analytics. Suppose everybody knew that generations were going to get bigger and then smaller and then bigger and then smaller. Would the stock market go up and then down and then up and then down, as I was saying, and as all these newspaper columnists were saying? Or would it just stay the same, as these less imaginative economist critiquers were saying, that it couldn’t have anything to do with demography? Because by rationality, people would look forward and they’d anticipate it going down, so they’d all sell before it went down, so it would never go down.
So it turns out that it does go up and down, and that argument the economists gave doesn’t make any sense, because they were implicitly assuming that the rate of interest stayed the same the whole time. And if the rate of interest doesn’t stay the same, then there’s no reason why the stock market has to stay the same. So let’s go back to where we were before and change the model a little bit.
Let’s suppose that the model is an alternation between big generations and small generations. So let’s suppose that you have (6, 2), (3, 1), and (6, 2). I think I kept–so it’s a good question. What should I do with land? I think I kept it at 1 every period. God, I hope that wasn’t the key to the result. So in the paper, of course, I had, you know, people living 80 years and I had their income reflecting the sort of trend in income over the lifetime. But anyway, for the purposes of this class, let’s say there’s a small generation where it goes young 3, old 1.
Then the next generation, which is just 2 guys, so they’re both (3, 1) guys, so in aggregate, they’re going to be (6, 2). The next generation’s (3, 1), the next generation’s (6, 2), the next generation’s (3, 1) and it keeps alternating like that forever. Now, how would you solve for equilibrium? Does the whole thing just get much more complicated? I mean, what happens, after all? Any questions here about what we’re doing?
So it looks way more complicated, but it’s not. It’s a little bit more complicated, but not much. You can still solve for this. So I’m going to not spend too much time on this, but I’m just going to show you what you would do. So what would you do? Well, by symmetry, I claim, you can guess that there are going to be two prices.
So you notice that every generation only cares about the relative price when it’s young and when it’s old. So there’s this relative price. That relative price between consumption at time 1 and time 2, that’s going to affect the generation born at time 1. It actually has no effect on any other generation. These generations are born after that, so this interest rate, this relative price between these two periods affects only 1 generation.
So this generation is affected by the interest rate between time 2 and time 3. This generation is affected by the interest rate at time 3 and time 4. Every generation is responding to the interest rates during its life, not during another generation’s life. So by symmetry, I think, you can guess–and it turns out to be true–that there’s going to be a price, a relative price–I should call them small–for the small generation, and a relative price for the big generation. The same thing is going to happen.
So this relative price will be psmall, this will be pbig. Then we’ll get psmall again here, and pbig connecting these guys. And it’s just going to keep repeating itself over and over again. And by symmetry, you could guess that the equilibrium would look like that. Is this making sense? It’s not making sense. Is it making sense? So what would I have to check to see that there’s equilibrium? Let’s do it over here.
Let’s do a small generation. Okay, so we never did the first. So the first generation, it depends whether we start with a small one or a big one. But remember, if we can clear all the markets for all time, except the first market, that one has to automatically clear. That was Walras’ insight. So we know that if we can clear all these, the first one’s going to look very different, but it’s going to clear anyway, so I don’t really have to worry about it. So we’ll see why that works. So I’m going to now talk about market clearing at any time, except the first one, which is more complicated.
The first one seems really complicated because there’s this guy 0 who owns these dividends that go on forever. But of course, they’re just worth the price of land, which he’s going to sell off. But anyway, we’ll come back to that in a second.
So in every generation, every time period, when small generation is young, what do you get? What’s the market clearing equation going to be? Well, the small generation is going to be young, so that means the old generation is big. Sorry, it’s a half-year, so the big old guys, their endowment is going to be 6. There’s 2 of them, so I can put the 2 out here. 3 + 1 pbig, divided by pbig, that’s the big old generation + and the young generation, they’re the usual 3 + 1 pyoung, divided by 1. A has to = now, how many endowments do we have? We’ve got 2 old guys each owning 1, so that’s 2 times 1. That’s their endowment, + 3, that’s the young guys, + the land dividend. So that’s 6.
Student: If you divide the second <> small?
Professor John Geanakoplos: So this should be psmall, not pyoung. psmall. No, I don’t, because the old guys–the big generation is old. The small generation is young. So the young generation, they’re consuming in their youth now, so they’re looking at their endowment. The price when they’re young is 1. So their young endowment is 3, their old endowment–their income’s 3 times the price to them of 1 when they’re young, plus the endowment of 1 when they’re old times psmall, divided by the price when they’re young, which they’re thinking of as 1, right?
They only care about the relative price between young and old. That relative price is determined by psmall. So I could call it pt and pt+1, but I know that ratio is going to be psmall. So really, it only depends on the ratio, so I might as well just use the ratio, psmall. This is a crucial idea that I’ve used 100 times and I think–I’ve used it now 3 or 4 times anyway—so, but I think some of you don’t quite know what the hell I’m talking about. So somebody ask me a question if they don’t know why this is right.
Whoever just asked it, maybe you still don’t know why it’s right. Who asked this question a minute ago? Yes. Do you understand how I got this? You do. Okay, but when the old guys–see, the old guys, which is the big generation, we’re clearing time–we’re looking at this time. So the total endowment is 3 for the young generation. Their endowment is 2, and the 1 unit of land. That’s where I got the total supply of 6.
So this generation when it’s young, they care about the psmall, the relative price between here and here, and so they’re looking at psmall, the relative price of 1 to psmall. The old generation at the same time is looking over their lifetime at the ratio of consumption here to here, which is given by pbig. So they’ve got a pbig here, and they’re consuming when old, so they’re dividing by pbig, okay?
But now we’ve got a second equation, which is when the big generation is young. So why don’t you tell me what that equation’s going to be? How much is the endowment when the big generation is young? That’s, maybe, clearing it this time. Here the big generation is young, so what’s the endowment? It’s going to be the young–so the old, there’s only going to be 1 old guy. There are going to be 2 young guys, each have endowment of 3, + 1 unit of land, so that’s 8. You can see that adds up to 6 + 1 + 1 is 8. Okay, and what’s their demand? What’s the demand for the 2–so there’s the old. What are the old going to be doing here? It’s in this generation we’re doing the market clearing. Somebody knows how to answer this. Yeah.
Professor John Geanakoplos: : Over here, the young, the small generation only has 1 when it’s old.
Student: Big generation.
Professor John Geanakoplos: : But big generation, that’s over here. We’re clearing at this time period, so there’s a bunch of young guys and just a few old guys. So the endowments are 6 + 1 + 1. The time is going that way, so that’s why that’s an 8 over here. So if this is the 1 old guy’s endowment, 2 young guys who have 3 each, that’s endowment of 6, + the land is 1, so you get 8. That’s how I got an 8 over there. That’s the supply. So what’s the demand? Who’s trying to buy the stuff? What’s the demand of the old guys?
Professor John Geanakoplos: And that’s going to be what? They’re planning to spend when they’re old. We’re doing this one. They make their plans when they’re young. They anticipate what the prices are going to be and so how much do they plan to spend when they’re old?
Student: 1 half times 3 + 1p–
Professor John Geanakoplos: 3 + 1p what? p what? What p?
Professor John Geanakoplos: psmall, exactly. Over psmall, plus
Professor John Geanakoplos: What are the young guys doing? We’re over here. So 2. There are 2 of them. What’s this? p what? Divided by? Oh, there’s 1 half here.
Professor John Geanakoplos: 1. Okay, exactly. So that’s it. So there are 2 equations and 2 unknowns, pbig and psmall, and 2 equations. So to solve this whole model, you could get 2 equations–you just have to solve these two simultaneous equations, which I did. And then you can find the effect. So the economist critiquers would say, “Well, pbigshould equal–” What would be the price of land, by the way? So what’s the price of land going to be in the end? What’s going to be the price of land, according to Fisher, at time 1? At time 1, let’s say. So right here, what’s the price of land? Remember, it’s the price after the dividend is paid, so what’s the price of land?
Professor John Geanakoplos: Let’s do the present value of the discounted dividends. What would it be? Maybe I’d better write it over here. Remember, the dividends are just 1 + 1–you know, 1 +–so there’s going to be 1 + 1 + 1 + 1, but I’m going to be multiplying by something. So what is the price of land? psmall, incidentally, =–I could always write that as 1 over 1 + rsmall and I could write pbig as 1 over 1 + rbig.
Okay, so what’s the price of land going to be? So it’s the price of land starting here just after the dividend is paid. So it’s hard to think about this, but if you can figure this out, then you’ve totally understood everything in the course, because this is so much more–it uses all the ideas in a just way more complicated way. Yes?
Professor John Geanakoplos: Well, start telling me the formula. So if I want–when I do the first terms, what is this first term?
Professor John Geanakoplos: So you would put 1 + rsmall here? Okay, and now what would you put over here?
Student: 1 + rbig.
Professor John Geanakoplos: You’d put 1 + rbig here?
Professor John Geanakoplos: Okay, very good. So now you guys really know what’s going on. What would you put over here?
Professor John Geanakoplos: Okay, exactly. 1 + rsmall time 1 + rbig, times 1 + rsmall. Okay, that’s that one. I’m going to stop with that. Too much. + dot, dot, dot. So you have the idea now. What’s the idea? You’re taking the present value of all the dividends.
So this dividend, you have to discount it by 1 + rsmall. This one, you have to discount it twice. First to bring it back here, that’s 1 + rbig, then to bring it back here, that’s another 1 + rsmall. So 1 + rbig, 1 + rsmall. The third dividend, you discount first at the small discount, then at the big discount, then at the small discount, just like that. You keep adding them. So why should the prices–what if you look at the price of land at time 2? Is it the same number as that? Is it the same formula?
Professor John Geanakoplos: It’s not the same formula, because at time 2 you’re starting here, and the first discount is pbig. So it’s going to be 1 over 1 + rbig + 1 over 1 + rbig times 1 + rsmall + 1 over 1 + rbig times 1 + rsmall times 1 + rbig + dot, dot, dot. So this isn’t the same as that.
It would only be the same as that if rbig was = to rsmall, if psmall was equal to pbig. So which one do you think is bigger? Do you think psmall is = to pbig, or one of them is bigger than the other, and why? That’s the answer to the critics’ puzzle.
So you see, you’ve taken something that appeared in the newspapers. It sounds like common sense. If there are all these middle-aged people like me, bidding up–you know, there are so many of us who want to hold stocks. We’re at the peak of our earning power. We’ve got tremendous amounts of money. What are we going to do with it except put it into stocks. You’d think common sense would be the price of stocks would have to go up.
When we get old and we’re selling all our stocks, you’d think the price of stocks would have to go down, okay? So that would tell you that the interest rate was going to be bigger or smaller. So that would suggest that the price at time 1, when the small generation was young, would be low, and the price at time 2 would be high. And that could happen if this interest rate was low, and this interest rate was high.
But does that happen or doesn’t that happen? You can’t know until you solve in the model. The critics said, “Oh, that can’t be right, because somehow the price should even out and everybody should be looking ahead and buying or selling at the right time.”
You can’t buy and sell at the right time, because you’re going to be dead. So you don’t have much choice, except when you’re young here to save, and when you’re old here to dis-save. They seem to have overlooked that small problem.
So the point is that, to know the answer, you can’t just reason it out verbally. You have to write down a model, solve for the model and see what happens. And sure enough, it happens–so we go back to the paper now.
Okay, are there any questions about this? Okay, so all I did in this spreadsheet is I wrote down these 2 equations. So you see that I’ve got the young and the old are the same half, half log utilities. The small generation is 3, 1, the big generation is 6, 2. Land is always producing 1 every period and we get 2 different prices. The price is called pA. That’s the A generation, which is the small generation and the B generation, which is the big generation.
So you can see that pA is a very small number when the generation is small, and that means the discount rate is very high. So a high interest rate when the generation is small, and a low interest rate when the generations are big. So in the money making peak of the big generation, the real interest rate is small, and so that means you’re not discounting very much, and so the prices are very high of land.
And so sure enough, the price when the small generation is young is 1.29, and when the big generation is young, it’s 2.08, vastly higher. So it really is the case that as the generations rise and fall, the interest rates rise and fall and the price of land rises and falls, and everybody’s rational, anticipating everything.
So it really could be–I’m not saying it’s true, because it could be a coincidence that these things all matched up, but the fact is, demography can have a big effect on economic equilibrium in the stock market. And you can be an economist and believe in demography. You don’t have to be a Marxist or something to think demography and fertility and all that stuff is going to affect the economy. For a long time, economics tried to wall itself off from other social sciences and say, “Oh, those fertility people and, you know, all those touchy-feely people, it’s got nothing to do with the real markets.” Well, it has a lot to do with the real markets. Okay, so one last thing, would you rather be in the small generation or the big generation?
Professor John Geanakoplos: Of course, the answer’s there, but tell me why. Yeah?
Professor John Geanakoplos: Okay, that’s correct. So I’m screwed. And so why is that?
Professor John Geanakoplos: It turned out to be the utility of the–this is the utility of the big generation per person. Of course, adding the two people together, it’s bigger, so per person it was this, a lot less than the generation of the utility of the small. So why did that happen? Why exactly is that happening? So remember, I’m distinguishing your peak. I’m calling young your peak earning years and old I’m calling the stuff when you’re retired, I’m not earning very much. So why is it that a high interest rate is good? Yes.
Student: It encourages consumption today for the younger generation.
Professor John Geanakoplos: Well, I think the point is, if you’re making money when you’re young, and you’ve got a high interest rate, that means a high return. So you can make a big return through your life, whereas my generation had a very low return. The return I’m getting now is terrible. It’s negative. The return I’m getting now is just hideous.
But when my parents were young, they were making an astronomical return, incredible rates of return. And they were earning money in their younger years and investing it to save when they were old and they were getting incredibly high return. There was nothing to do with my money. There are so many people like me pouring money into the stock market, we’re getting a very bad return, and that’s why it’s not so good to be part of a big generation.
So other people had said the same thing. They said big generations have to squeeze into the same houses, so there’s less housing stock, or we build more houses, the next generation comes along, there are all these houses built for them and there are less of them, so they get better houses. Everything’s better when you’re the small generation. You’re screwed as the big generation and here’s an example of it.
Chapter 4. Philosophical and Statistical Framework of Uncertainty [01:02:30]
Okay, so that’s it for Social Security. All the things I wanted to say about Social Security. Now the exam, as I say, you won’t have to do the two generations and all that stuff on the exam. But the first part about solving for the .55, that one you might have to do something like that.
But anyway, the lesson that you can have a controversy in the newspapers with arguments on both sides, you never can figure out the right answer till you write down a model. And the model can be very simple, the kind of thing you could do as an undergraduate. That’s the lesson I’m trying to communicate. All right, so from now on though, we have to change gears and I think the course gets a lot more interesting, okay. But we needed all this background basics to have the logic of finance.
But finance is really nothing without uncertainty. If you don’t know what’s going to happen tomorrow, that’s when the thing gets interesting, that’s when you have to think harder, that’s when people make mistakes, that’s when a hedge fund can make a profit, that’s when everything gets more exciting, when there’s uncertainty, and we have to start adding uncertainty to our thinking about the world.
So since I only have a few minutes, I’m going to spend the last few minutes just reminding you of what you’re supposed to know about uncertainty, which is this slide, okay? So it’s not a very complicated thing. Where’s the–I see PowerPoint. Okay so there are two kinds of uncertainty, one of which we’re never going to get to.
So the uncertainty that we’re going to work with in this course is everybody understands what could possibly happen. So this was a brilliant states of the world model, it’s called. So Leibniz is supposed–see, it says states of the world right there. Leibniz was the one who first invented the idea of states of the world. He said, “We live in the best of all possible worlds.” So if you don’t know what’s going to happen in the future, we’re going to represent that as many potential states of the world.
Now there are other ways you could model uncertainty, and it’s a big subject in epistemology and philosophy. I’m going to stay–hew closely to the states of the world thing. So if there’s something about the world we don’t know tomorrow, I’m not going to assume that it’s just vague in our minds and we can’t quite make it out. I’m going to assume you actually can make it out. It’s just you don’t know which of the possibilities are going to happen. So it’s many states of the world.
So an example of that, of course, is spinning a dice, or spinning two dice, so you don’t know which of 36 possibilities is going to happen. Each of those would be one state of the world.
Okay, so those are all the possible sums, so there are more states of the world than there are outcomes, notice. You can get 2 through 12, is the sum of the dice, but there’s still 36 possible states of the world. And so that random variable, the sum of the two dice, is expressed as a function. It says, in every state of the world, what is the random variable’s value?
So our state of the world, by that I mean something that’s so detailed that it tells you the value of every random variable. So that’s how I’m going to be thinking of uncertainty, that we’re here today and a bunch of different things could happen tomorrow, and this is an exhaustive list of everything that can happen tomorrow. We’re going to have to come back and criticize this assumption, but it introduces uncertainty in a way which is still quantifiable and not vague. So you can talk about the histograms of outcomes of the dice, you know, how many ways, what are the chances of getting a 7, 6 out of 36, etc. So this is stuff I’m sure you all know, okay.
The most famous histogram of mathematics is the normal distribution, which I don’t have time to explain. And so a normal distribution, remember, is this exponential thing. The dice example, you know, sort of looks like it. In fact, if you had enough dice, and you kept averaging their outcome, it would be normally distributed. So lots of things turn out to be normal. One of the properties of normal is that the tail gets small very quickly, and so you’re not going to get very many outliers.
The normal distribution is a very common distribution. There’re lots of reasons why it should occur all the time, and it’s also mathematically very easy to work with. So economists for a long time assumed that everything was normally distributed, but an implication, as I say, of normal distribution is that it’s very unlikely something extreme is going to happen, because this thing goes down exponentially fast to 0, so you’re almost never going to see something way out here.
Yet we get these crashes fairly frequently. Every 10, 20, 30 years or something, there’s some gigantic outlier. So clearly things can’t be normally distributed, and that’s called fat tails. I’m sure you’ve heard about that in the newspaper. We’re going to come back to that.
Now there are a couple of things that you need to–so you can represent the payoff of any random variable as a picture, and every state, what is its payoff? So a riskless random variable, riskless investment, will pay the same amount in every state. I still have a few minutes here.
Okay, so now two incredibly important concepts, which I’m sure you know–or at least the mean you know–just the average, what’s the expected payoff? So the expected payoff is, you multiply the outcome in every state by the probability of the state and you add it up. Okay, so there are how many different ways? So if you have 1 dice, you can figure out the expectation of 1 dice. It’s equally likely to get 1, 2, 3, 4, 5 or 6, and so if you add all those up, multiplied by a sixth, you get 3 and 1 half. So the expectation of 1 dice is 3 and 1 half.
The expectation of two dice is therefore–okay, that’s 1 dice. Okay, now the expectation of adding 2 dice together is of course going to be 7, because the expectation of 1 dice is 3 and 1 half and the other dice is 3 and 1 half, and so the expectation of the sum is the sum of the expectation, so it’s 7. Okay, so you know that. Now one more thing that you have to know is the variance.
So how do you measure how uncertain you are about what’s going to happen? Well, the simplest number to do, like with the dice, is to say, the expectation is 3 and 1 half. When we don’t get 3 and 1 half, it means there is some error. Expectation wasn’t confirmed. So you could just measure the error as the distance between 3 and 1 half and what you actually got. So if you got a 4 you could say you were off by 1 half. But people don’t do that. They measure the square of the distance and they take the average square of the distance, and that’s called the variance.
So if you got 4 when you were expecting 3 and 1 half, the error is 1 half squared, so you take 1 sixth times 1 half squared, + 1 sixth times 1 and 1 half squared, + 1 sixth times 2 and 1 half squared. That’s for the high mistakes. And the low mistakes, you’d take 1 half times 1 half squared, + 1 half times 1 and 1 half squared, plus 1 half time 2 and 1 half squared [correction: 1 sixth times each error, not 1 half].
Now why do they use the expected squared error instead of just the expected error? Well, for this simple reason, which you may not have thought of: if you had to guess–if you looked at the error as the penalty for making a wrong guess, and you were forced at the beginning to say, “What guess would you like to make for how the dice is going to turn out–how the dice is going to turn out and if you’re wrong, we’ll penalize you by this measure of error?”
Well, if you were just taking the absolute error, your best guess would be any number between 3 and 4. You could say 3 and 1 half, but you could also say 3 and 1 quarter, because if you said 3 and 1 quarter, you’d become closer on all the low rolls of the dice, and be a little further off on all the high rolls, and you’d get exactly the same error. So you could say any number between 3 and 4.
But if you make the error the square of the mistake, your best guess is always to say the expectation. 3 and 1 half is the only thing you can say to minimize your expected penalty, if the penalty is the squared error. So because of that–so that’s connected to a mathematical thing and orthogonal vectors and all that. Because of that, the error we talk about always is the squared error.
And so that’s whenever we’re measuring uncertainty, we’re going to talk about the squared error, the average of the squared errors–or the square root of that, which is called the standard deviation. Okay, I’ve one more minute to just finish this. I’m assuming you all know that or can learn that.
One more thing, I’m going to assume you know, the last thing is the covariance. What does it mean to say that random variables X and Y move in the same direction? So this was quite a brilliant thought. So what it means is that to move in the same direction, means if X surprises you by being above what you would have expected, so if the expectation of X is X bar, and the expectation of Y is Y bar, if you multiply the 2 and they’re both higher than you expected, you’ll get a positive number. If they’re both lower than expected, you’ll also get a positive number.
So by taking this sum, the expected deviation from the expectation, multiplied, you get a number which, when it’s very positive, it means they’re either both positive together or both negative together. So they’re going in the same direction. If one of them is above the average while the other one’s below the average, then you’ll get a negative number. So big covariance means they move in the same direction. Big negative covariance means they move in the negative direction. Okay, so I’m going to take it for granted you know this, and we’re going to take this up, starting right after the exam, which is on Tuesday.
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