## 6. Irving Fisher’s Impatience Theory of Interest

Prof: 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

X_1 1 half log X_2 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 U^(A)

and U^(B) the same as before, and E-hat^(A)_1,

the endowments, E-hat^(A)_2 and

(E-hat^(B)_1 E-hat^(B)_2).

So we’ve left out half the

variables up there and we define E-hat^(A)_1=

E^(A)_1=1 and E-hat^(B)_1=

E^(B)_1=1, but E-hat^(A)_2 (this

is the Fisher insight)=E^(A)_2 what A owns of

the payoffs of the future dividends,

[theta-bar^(A)_alpha times D^(alpha)_2 plus

theta-bar^(A)_beta times]

D^(beta)_2. 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-hat^(B)_2 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-hat^(A)_2=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

q_1–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 P_1=1,

P_2=a third, and sure enough

X^(A)_1=4 thirds, X^(A)_2=2,

and X^(B)_1=2 thirds,

and X^(B)_2=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. 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

pi_alpha– (this is a big

pi)–pi_alpha=a third,

so Fisher says well, you don’t have to solve for the

whole equilibrium to figure out what pi_beta is.

What would pi_beta be?

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 pi_alpha is

1 third, pi_beta 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 q_1=q_2=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

X_2 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 X_1 is 1 dollar today,

right? If you had one apple today you

could sell it for q_1 times 1 apple,

which is q_1 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 X_2. So 1 unit of X_1

today turns into 3 units of X_2,

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

q_1=1, q_2=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 q_1=1

and q_2=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 pi_alpha

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 pi_alpha?

So if I re-solved equilibrium

taking q_1=1 and q_2=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 pi_alpha

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? Prof: Price of stock

alpha. What was it before?

Student:

>Prof: So what was it

before? Student: 1 third.

Prof: 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 P_1.

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

P_2 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 P_1=1

that’s what Fisher would say. Assuming P_1 is 1 you

figure out how many units of today’s good is it worth.

Now, if P_1 isn’t 1

then what do you do? Suppose P_1 were 5

and P_2 were–or P_1 is 6,

let’s say, and P_2 is 2 then what would you do?

You’d have to say P_1

times pi_alpha=D^(alpha)_2.

So if you multiplied all the

prices by–am I putting the P_1 down at the bottom

or the top? If you multiply out all the

prices by 3, just leave it like this.

We’ll say if pi_alpha

=P_2 times D^(alpha)_2.

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 P_1 divided by

P_2 (if P_1 is 1,

assuming P_1 is 1) times D^(alpha)_2,

which is 1 over (1 r) times D^(alpha)_2.

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 P_1 over

P_2 is just 1 r, 1 r is P_1 over

P_2. 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 P_2 is only a third of

P_1. 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.

P_2 is the amount you

give up today to get an apple next year, so it’s a third.

Another way of saying that,

if P_1 is 1, is that the real interest rate,

the tradeoff between apples tomorrow and apples today which

is P_1 over P_2,

because 1 apple today can give you three apples tomorrow,

so P_1 over P_2 is 3,

so 1 r is three. So the apple tomorrow is worth

P_2 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. So another way of saying the

same thing is you could turn cash next year into cash this

year. So assuming q_1 is 1,

another way of saying that is 1 over 1 i times

D^(alpha)_2 times q_2.

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 P^(alpha) was still a third?

Because the real interest rate

hasn’t changed, it’s still 200 percent.

So D^(alpha)_2 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. Prof: So that’s right,

now how did he do that? Because let’s just do it.

You take 1 dollar today at

price q_1=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 X_2,

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.

Prof: Usually,

and do you think it’s higher or lower than usual now-a-days?

Student: It’s probably

lower. Prof: 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.

Prof: Who said that?

That’s a good–1 percent,

that’s about right. So what is the real rate of

interest now? Student:

>Prof: What?

Student:

>Prof: 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.

Prof: 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 q_1=1 and q_2=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?

Student: No.

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-hat^(A)_1,

let’s call it X-tilde ^(A)_1,

X-tilde^(A)_2, and Xtilde^(B)_1,

Xtilde^(B)_2 that made everyone better off.

Then, well, we give their old

proof. Then what?

It means that P_1

Xtilde^(A)_1 P_2

Xtilde^(A)_2 is bigger than what?

P_1^(

)E-hat^(A)_1– (all right, that’s what you

have in the Fisher economy) P_2 E-hat^(A)2 and

similarly P_1 X-tilde^(B)_1

P_2 X-tilde^(B)_2 is

bigger than P_1 E-hat^(B)_1

P_2 E-hat^(B)_2.

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.

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. Prof: 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?

Student: Yeah.

Prof: 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? Prof: 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. Prof: 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? Prof: Yes,

I can raise which board, not this one?

Student: Yes, that one.

Prof: Yeah.

Well, sorry.

Student: Oh.

Prof: 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.

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 P_1 and P_2

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.

Prof: So why does it go

up? That’s correct.

Student: Because there

needs to be more of an incentive to save.

Prof: 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 P_2

(let’s say)=1 and we’re solving for P_1 and

here’s the supply, this is X_1,

and here’s demand. So remember X^(A)_1

is going to be something like P_1 E^(A)_1

P_2 E^(A)_2 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 P_1.

So if P_2 is 1 then

this is just equal to 1 over 1 delta times (E^(A)_1 1

over P_1 times E^(A)_2).

So clearly the demand goes down

as P_1 goes– as P_2–this is

P_1, so P_2=1,

so if I divide by P_1,

P_1 over P_1 goes away.

Then I have P_2 over

P_1, and if P_2 is 1

that’s just 1 over P_1.

So obviously as P_1

goes up your demand goes down. That’s just what you’d expect.

So P_1 goes down the

demand goes up, or P_1 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 A_1 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 X^(A)_1 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 P_1.

Implies P_1 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 P_1. But if you raise P_1

holding P_2 fixed that’s just P_1 over

P_2, 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 E^(i)_2?

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.

Prof: It’ll what?

Student: Decrease.

Prof: Decrease, why?

Student: Because you’re

expecting things to be better>

signifies people will save

less. Prof: To save less or to

save more? So let’s think of good

X_1. If people thought they were

going to be richer at the old prices what would they do today

for X_1 demand more or less today?

Student: The rate would

go up, right? Prof: Yeah,

the right answer is up. He said down,

but let’s just figure out why. Student: They demand

more>Prof: 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

X_1? 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 P_1 relative to P_2 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. Prof: That’s going to

lower it. Why is that?

Student:

>Prof: 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? Student: Down.

Prof: 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. 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.