Claire Corlett

Fish Food, Fish Tanks, and More
6. Irving Fisher’s Impatience Theory of Interest

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.

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