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PROFESSOR: Our topics part of
the course by revisiting
another important topic that--
one of the most important topics
in economics which is
where does capital come from.
This is a topic that is
essential in economics.
It's also really central, the
basis, for what is in finance.
So a lot of what we'll
do today is really--
if you want to learn more about
it you take more in
economics but also more
courses in Course 15.
So, basically, we spent a lot of
time this semester talking
about one input to
the production
function which is labor.
We talked about labor supply,
labor demand, monopsony
models, et cetera.
We haven't talked much about
the other input into the
production function
which is capital.
Now, partly, that's because
that's a more awkward concept.
It's clear what labor is.
Labor is the workers working
in the production process.
Capital's a little bit harder
because capital's sort of
everything else-- the machine,
the land, the buildings, other
physical inputs.
And we know where labor
comes from.
Labor comes from our working.
But it's less clear where
capital comes from in some
aggregate concept.
So, basically, the key thing is
that all forms of capital
have a common feature.
All forms of capital have a
common feature which is what
capital represents is a
diversion of current
consumption towards future
production and consumption.
So capital's about diverting
current consumption towards
future production
and consumption.
So the original concept of
capital came from farming
where the notion was that
farmers every year would take
some of their grain, and, rather
than eating it, they'd
put it aside to become seed to
plant for the next year.
That was their capital.
And so they diverted their
consumption this year which
was eating the grain they grew
to produce future consumption
through planting those
seeds and creating
consumption for next year.
So, basically, in a modern
economy the idea is the same.
And so when we think about
capital, what I want you to
think about is I want you to
think about, basically, the
capital as money.
Think of the capital in the
production function as the
money that we invest in all
these other things that aren't
labor-- that we invest in
machines, and we invest in
buildings, and we
invest in land.
So we want to think about
capital not as physical
capital but as financial
capital.
That's the way to be thinking
about that one aggregate
letter k is this financial
capital.
It's the money that's invested
in producing goods, in
building machines, and
building buildings,
and stuff like that.
OK?
Now, basically, where do
firms get that money?
So you're a firm.
You want to build a building
or new machine or
something like that.
Where do firms get that money?
They get that money in
capital markets.
Capital markets are basically
pools of money that firms draw
on to invest and
create capital.
So a capital market--
literally think of it as a pool
of money that's out there
that firms can tap into if they
want to build a building
or build a machine
or buy some land.
They tap into capital markets
to make those investments.
So while capital physically
represents lots of different
things, financially it
represents one thing which is
the pool of money that firms tap
into to invest, to divert
current consumption to
future consumption.
OK?
It's the pool of money firms
tap into to invest. That's
what we mean by capital.
By capital market we represent
that pool of money.
So think of capital as financial
capital, and think
of where you get financial
capital in a capital market as
being that pool of money that
firms tap into to make
investments to divert
towards the future.
Now, where does that supply
of money come from?
Where does that supply
of money come from?
Well it comes from households'
decisions on how
much to save. OK?
So the pool of money in capital
markets, the pool of
money that firms draw on to
build capital, comes from
households' decisions
on how much to save.
So now we see the tie to labor,
the other input in the
production function.
Just as households' decisions on
how hard to work determines
the labor input into the
production function,
households' decisions on how
much to save determines the
capital input into the
production function.
So just as my decision how much
to work determines how
much labor is available to
firms, my decision on how much
to save that's what fills
up this pool.
OK?
So this pool of financial
capital is filled up by
household savings and then drawn
down by firms' demand
for investment.
And that's the way a capital
market works.
So, basically, if we think
about capital market
equilibrium--
if you go to Figure 21-1--
we have equilibrium in
capital markets.
Now this is just like we talked
about-- this is the
other factor markets.
Just like we talked about labor
markets and determining
what determines the wage rate
and the optimum amount of
labor hired, it's same
with capital markets.
You have some demand
for capital.
That comes from firms' demand
for investment.
Firms want new machines.
They want new buildings.
That's downward sloping because,
initially, there's
very high demand for capital.
But there's a marginal
diminishing product.
The more capital I have the
less valuable it is on the
margin, the less you are
willing to pay for it.
So there's a downward sloping
demand curve for capital and
an upward sloping
supply curve.
And the price, in this market,
is the interest rate.
What is the interest rate?
The interest rate is the rate
you have to pay households to
get them to lend you money.
So the interest rate, i, is
the rate you have to pay
households to get them
to lend you money.
So if that interest rate is
very high, firms will not
demand much investment because
they'll have to pay a lot of
money to get the financial
capital to finance that
investment.
But households will be delighted
to supply lots of
savings because they're
getting paid a
high price for it.
So, basically, the interest
rate serves as the
equilibrating price
in this market.
Just as the wage serves as the
equilibrating price in the
labor market, the interest
rate serves as the
equilibrating price in
the capital market.
As the interest rate rises,
folks want to save more,
filling more money into
that pool of capital.
And firms want to borrow less,
taking less money out of that
pool of capital.
And when that supply and demand
is equilibrated, at
point e, is going to be where
the firm's drawing on the pool
at exactly the rate people are
putting money into the pool.
And that's going to be
the equilibrium.
OK, so we want to focus on--
for today's lecture and
next lecture as well--
is what determines the money
that goes into that pool.
We know what determines the rate
at which firms want to
draw out of that pool.
That's basically going to be
determined by the production
function and all the stuff
we learned in lectures on
production theory.
You can get your optimal
demand for capital.
It's going to be determined
by isocosts and isoquants.
And you get some k star.
But what's going to determine
what goes into that pool?
That's going to be households'
decisions to save. And
households' decisions to save,
we say, are determined by a
process we call intertemporal
choice.
Added some extra letters there--
intertemporal choice.
Intertemporal choice--
which is basically, instead
of thinking about someone
choosing between apples and
bananas, we think of them
choosing between consumption
today
and consumption tomorrow.
So think of different periods
like different goods.
And I'm choosing between
consumption today and
consumption tomorrow.
That's my intertemporal
choice--
the rate at which I choose to
trade off consumption in
different periods.
So for example, I'll illustrate
how this works.
Let's say that I'm deciding
whether to just tell MIT,
"Look, I don't want
to work next year.
I want to stay home and
take care of my kids.
You're not going to pay me.
I'm taking an unpaid leave for
a year." Something professors
can do with enough advance
warning to their
chairman and such.
So I'm going to take an unpaid
leave next year.
I'm thinking about doing that.
And now I have to
say, OK, fine.
Next year I'm going to take this
unpaid leave so I have to
decide how to allocate my--
and then I'm going to come
back and life will be
the same thereafter.
So it's just about next year I'm
going to take this unpaid
leave. I have to decide how to
allocate my consumption across
this year while I'm working and
next year while I'm taking
an unpaid leave.
And let's say my salary
is $80,000 a year.
So one thing I could do is I can
consume all $80,000 this
year and consume nothing
next year.
That would not be a very
satisfactory outcome as I die.
That's obviously not going to
be a satisfactory outcome.
But what's the alternative?
MIT's paying me this year,
and they're not
paying me next year.
What's the alternative?
Well, the alternative is I can
save. And by saving, what we
mean is I can loan some of the
money that I make out to firms
to invest in their physical
capital in return for which
they'll give me interest. And
next year I can live on the
interest I've earned from
making that loan.
Now, I don't literally go to
Genzyme and Microsoft and
Apple and say I want to
loan you money and
negotiate with them.
That obviously would
be impossible.
What I do is I implicitly loan
to firms through drawing on
various aspects of the
capital market.
So does anyone know, how can I
implicitly loan to a firm?
Let's say I want to--
Yeah?
AUDIENCE: Banks.
PROFESSOR: Banks.
So explain what you mean.
AUDIENCE: You deposit
money in a savings
account for the bank.
The bank pays you some interest
rate so that it can
use your money to loan to bigger
companies that want to
take money out of the bank.
And they, in return, get money
from the other companies by
the companies paying
some interest on
what they took out.
PROFESSOR: Exactly.
We call banks financial
intermediaries.
What that means is they
basically are the folks who
can get a hold of firms
and make those loans.
So, in other words, I don't
loan directly to Genzyme.
I loan to the bank--
Citizens Bank, my bank,
and Citizens
Bank loans to Genzyme.
Citizens Bank pays me an
interest rate on my savings--
now close to zero, we'll come
to that-- but basically pays
me some interest rate.
Genzyme pays them an interest
rate to borrow money, higher
than what they're paying
me, and the
difference is bank profit.
So, basically, one way I can
loan money is I can put in the
bank and get paid interest. we
don't think about putting in
the bank as a loan, but that's
basically what you're doing.
You're loaning it to the bank.
And they're paying you interest
rate, i for that loan.
What else can you do?
How else can you--
Yeah?
AUDIENCE: You can
purchase stocks.
PROFESSOR: You could
purchase stocks.
So, in other words, what I could
do is I could directly
go to a public company, and I
could take some of my $80,000
and buy stock in that company.
There I'm essentially directly
loaning to them.
I'm directly giving
them money.
Now, it's not a loan that's paid
back like a bank loan.
It's loan that's paid back
hopefully with my stock
becoming more valuable
or with a dividend.
So how can I loan?
One is I can invest--
I could put it in the bank.
The other is I can buy stock.
I can put it in the bank, and
the bank pays me interest. I
could buy stock, and that stock
pays off in two ways.
One is many companies pay what
we call a dividend, what is
called the dividend, which is
a quarterly payment that
companies make to their
shareholders.
So if I invest in a company
that pays a dividend, then
I'll be getting a quarterly
check from that company that's
a portion of my investment.
The other is what we call a
capital gain which is the
stock could go up in value.
So next year, if the stock goes
up, if the stock market
moves steadily-- it doesn't,
it jumps up and down, we'll
come to that--
but if it went steadily up I
could just sell some of that
stock next year and
have extra money.
So that's the other thing I
could do with my $80,000.
I could loan to a company
by buying their stock.
How else can I loan
to a company?
Yeah?
AUDIENCE: I don't know exactly
what the difference is, but
couldn't you also invest
your money in a
mutual fund or something?
PROFESSOR: A mutual fund--
that's a good point.
That would be loaning--
a mutual fund is essentially
loaning money to an aggregate
collection of companies.
So there are very different
ways I can do stock.
I can do a mutual fund, I can
buy individual stocks.
There's lots of different
ways, but those are all
different ways to buy stock.
Yeah?
AUDIENCE: You could buy bonds.
PROFESSOR: You could
buy bonds.
You could buy company bonds
which is I literally loan
directly to the company.
Stocks aren't really a loan.
I'm literally buying an
ownership share in the
company, and they're
paying me back.
I could buy corporate bonds.
I could buy corporate bonds, and
the way those work is it's
literally cutting out
the middleman.
I don't loan to Citizens Bank,
and they loan to the company.
I just loan to the company,
and they pay me back.
That's a corporate bond.
I can also buy, by the
way, I can also buy
a government bond.
You may know the government's
running more than a trillion
dollar deficit right now.
Somebody's got to
finance that.
So you can loan to the
government and get paid back
by the government.
OK, let's put the government
aside for a minute.
Let's focus where we just--
we haven't really
had a government
sector in our models.
We're where it's just you
and the companies.
So let's leave the government
channel aside.
But the other thing I can do
with my money is I can loan it
through bonds.
The point is that $80,000--
yeah, I'm sorry.
AUDIENCE: With stocks, if I
were to buy a stock from a
company, wouldn't it be
primarily and usually through
the secondary market?
I wouldn't be giving any
money to the company.
I'd just be giving money to
the previous stockholder.
PROFESSOR: That's true.
It basically depends on whether
the marginal stock
comes from a new issuance as
stock by the company or
through stock that's already
floating around
the secondary market.
That's a good point.
So in some sense the--
likewise with bonds.
A lot of bonds are traded
in a secondary market.
So I'm thinking about a simple
model where basically new
stock gets issued by the
company, I buy it.
More technically you're right.
It's just trading among people,
but that sort of makes
things complicated.
Let's put that aside for now.
So, basically, the point
is is my $80,000--
there's lots of things
I can do with it.
All of them yield
me some rate--
the key point is all of them
have the feature that I'm
diverting today's consumption
for tomorrow's consumption.
I'm taking some of my money
and, rather than eating it
this year when I'm working,
I'm loaning it out in some
way, shape, or form and getting
payback in next year
when I'm not working.
And we can summarize.
Now this is a very complicated
set of mechanisms, not to
mention the secondary
market issues.
And this is basically a
semester of 15.401.
This is basically a semester
of finance theory.
But basically what we're going
to do is compress this all
down and say that I get some
interest rate on my money.
However I do it, let's just say
that somehow I divert my
money through one of these
mechanisms, and it yields some
effective interest rate, i.
And you can know behind that
there's lots of ways I can get
that interest. But for now just
simplify it down and say
the main thing is I'm diverting
my consumption now,
and it's yielding some interest
earnings on that
diverted consumption, i.
So what that means is that for
every dollar I divert, I get 1
plus i dollars the next year.
So for every dollar of
consumption I divert, in one
of these forms, I get 1 plus
i dollars next year.
So, basically, I could
literally--
if I wanted to-- let's say the
interest rate was 10%, just
for example.
Now what that means is instead
of consuming $80,000 this year
and nothing next year, I could
consume nothing this year and
$88,000 next year.
Obviously, that's not very
satisfactory either.
So how do we think about that?
We think about that in
Figure 21-2 shows--
now this is a complicated
diagram we gotta
use to figure 21-2--
this shows the intertemporal
choice model, intertemporal
substitution we also call it.
So the deal is that now instead
of the x-axis being
pizza and the y-axis being
movies or all the other wacky
things we've done, now the
x-axis is first period
consumption.
The y-axis is second
period consumption.
You might say what's a period?
Well a period's whatever I want
it to be-- a day, a year,
10 years, whatever.
Sometimes I'll say a year.
Sometimes I'll say a period,
but the point
is it doesn't matter.
It's about the trade-off.
So in my example, c1 is
consumption this year.
c2's consumption next year.
And my trade-off is I can
consume $80,000 this year, or,
given the interest rate, I can
consume $88,000 next year.
Now the trade-off-- that's
a typo, by the way.
That should be minus 1.1.
OK?
This is a 10% interest rate.
The key point is the trade-off
is that, basically, I can
trade off for every dollar I
don't consume this year, I
consume 1 plus i--
1 plus r there, should
be 1 plus i dollars--
next year.
We use r and i interchangeably
for the interest rate, so 1
plus i, 1 plus r dollars
next year.
So, basically, what does the
interest rate represent?
This is important.
The wage rate I defined as
the price of leisure.
Remember what the
wage rate was?
It was the price of leisure,
that basically by working I
forgoed the ability to--
I'm sorry, by taking leisure
I forgoed the ability to
earn a wage, w.
So, literally, that was a price
of sitting around on the
couch was the wage, w,
I could've earned.
Likewise, the interest rate is
the price of first period
consumption.
By consuming today, I'm forgoing
the fact that I
could've earned the interest on
that money had I consumed
it tomorrow or next year.
So the interest rate is the
price of first period
consumption, just as the wage
is the price of leisure.
Yeah?
AUDIENCE: You said before
that r and i are used
interchangeably for interest.
Does that play into the cost
function at all?
Is the cost for capital going
to be the interest rate?
PROFESSOR: I'm going
to come to that.
That's exactly what I'll talk
about next lecture.
So, basically, this is the key
thing, but the key thing to
understand intertemporal
choice-- and the other
important point to understand on
why it's a bit harder than
labor is there's an extra--
well it's not harder.
It's the same thing.
Remember, we said we don't model
bads in this course.
We model goods.
So we're modeling your choice
of how hard to work.
We model the trade-off between
consumption and leisure.
And then we said define labor
as the total amount of hours
available minus leisure.
Same thing here.
We don't model savings.
That's a bad.
Now you might not think
[? some of these ?]
things are good, but
savings really by
itself is not a good.
Unless you're Scrooge McDuck--
does anyone know who
Scrooge McDuck is?
Wow, that hurts.
OK, he was this old cartoon
character when I was a kid who
used to, like, fill a
swimming pool with
money and swim in it.
Basically, unless you're like
that, the savings itself does
not give you utility.
We don't have savings entering
utility functions.
We have consumption entering
utility functions.
Savings is a bad.
Savings is the mean by which
you translate consumption
period one into consumption
period two.
But from the effect of today
you wish you didn't have to
save. You just do it because
you want to make
sure you eat tomorrow.
So we model the good.
The good is consumption in
period one, and savings is the
difference between income and
consumption in period one.
So we don't model savings.
We define savings
as y minus c1.
We model c1, and define
savings as y minus c1.
You can see that there
in the diagram.
Now what happens when the
interest rate changes?
Let's go to Figure 21-3.
What happens when the interest
rate changes?
Actually, go to 21-4.
OK?
Skip 21-3.
Got to 21-4.
What happens when the interest
rate changes?
So, initially, we're at a point
like a and then the
interest rate goes
up from r to r2.
The interest rate goes up.
Now what does that do?
Well, graphically, it steepens
the budget constraint.
What that means is it's raised
the opportunity cost of first
period consumption.
First period consumption is now
effectively more expensive
because I'm forgoing a better
savings rate by eating today.
The more of my $80,000 I consume
today, the less I get
to save for tomorrow.
And that's now a better deal to
save for tomorrow because
I'm getting a higher interest
rate on that.
So what does that do?
Well that has two effects.
Just like a change in the wage
rate has two effects-- a
substitution effect and
an income effect.
The substitution effect, which
we can unambiguously sign, is
the fact that now first period
consumption's gotten more
expensive, so we
do less of it.
Substitution effects are always
price goes up, you do
less of the activity.
The substitution effect is first
period consumption's
gotten more expensive, there'll
be less of it.
Now here, once again, don't
slip into thinking about
savings yet.
You'll really get yourself
in trouble, and
it's a natural tendency.
Model consumption that makes
savings a residual.
So the activity we're modeling
here is first period
consumption.
The price of first period
consumption's gone up, so you
do less of it.
That's the substitution
effect.
The income effect is
you're now richer.
And you might say what do
you mean I'm richer?
I still have the same
$80,000 in income.
But any given dollar of savings
yields more income in
the second period.
So, overall, you're richer.
If you take the perspective of
saying I have two periods in
this model-- first and
second period.
Any given level of savings
makes me richer
in the second period.
That means I'm richer.
If I'm richer, I consume more of
everything including first
period consumption.
So first period consumption
goes up
from the income effect.
I find this confusing.
I don't know if you guys do, but
once again-- run through
this again.
I'm richer because for any given
amount of savings I now
have a total larger sum of
money over both periods.
When I'm richer I consume
more of everything.
One of the things that I consume
more of is first
period consumption.
So, actually, first period
consumption goes
up, and I save less.
It's sort of bizarre.
Because I'm getting
more return to my
savings I save less.
Here is the way I like
to think of the
intuition to make it easier.
The way I like to think of the
intuition is imagine that you
have a goal for saving--
something I call a target
savings level.
Imagine you said, look.
Imagine you said that I really
want to make sure that I have
certain level of savings
to live on next year.
Well if the interest rate goes
up, I can save less to get to
that target.
Right?
If I have a target, c2, and the
interest rate is higher I
can consume more c1 and still
hit my target of c2.
So that's the income effect.
I can effectively consume more
c1 because I'm made richer
because any given level of
savings allows me to consume
more the next period.
Now the target's an extreme
case, but I find it a useful
intuition for thinking about
what's going on.
And that's the income effect.
Now, obviously, as with
anything, this is ambiguous.
If you go to Figure 21-3 now
here's a case where the income
effect dominates.
Well, actually, go
back to 21-4.
Let's finish this example.
So here with the substitution
effect dominating, when the
interest rate goes up I consume
less in period one
which means I save more.
And that was prior intuition.
A higher interest rate
means you save more.
But we'll work through the
mechanics of how we get there.
And the reason the mechanics
is important is because of
Figure 21-3.
Which as in 21-3, the interest
rate goes up, but I save less.
The interest rate goes up, but
I consume more in period one
and therefore save less.
And that's consistent with
this target notion.
That basically I'm so--
All I care about-- let's say
in the limit if all I care
about in the limit is exactly
what I consume the second
period, then, basically, my
period one's consumption will
definitely go up from a raise in
the interest rate because,
basically, I'm saying, look, all
I care about is what I get
in the second period.
Now I can save less and
get to that target.
So my consumption first
period goes up.
The income effect dominates.
So the bottom line is, just
like with labor supply, we
can't tell.
Unlike with goods where it's
rare to see a Giffen good,
here we honestly don't know
whether a raise in the
interest rates will raise
savings or lower savings.
It all depends on the
strength of the
substitution and income effects.
Let me actually say one of the
most disturbing things in
empirical economics is we
actually do have no idea.
Literally, there's no convincing
study out there
which even tells us which way
the effect of interest rates
goes on savings.
We think probably the
substitution effect dominates,
but it's been very
hard to find a
convincing estimate of that.
So it's a little bit disturbing
for empirical economics.
We'll typically assume it
dominates, but don't
necessarily assume that
in the real world.
OK, questions about that--
intertemporal choice
framework?
OK, now, with that in mind,
let's now talk about how
capital markets work.
How do capital markets work?
And the key concept for thinking
about capital markets
is the concept of
present value.
And the concept of the present
value is simple.
It's that $1 tomorrow is worth
less than $1 today.
$1 tomorrow is worth
less than $1 today.
And why is that?
It's because if I had the
dollar today, I could've
invested it in something
productive and had 1 plus i
dollars tomorrow.
So if you give me a dollar today
I could have 1 plus i
dollars tomorrow.
If you give it to me tomorrow,
I just have $1.
So by definition, $1 today is
worth more because I have the
opportunity to save it.
Whereas a $1 tomorrow
I don't have the
opportunity to save it.
It's too late.
So the key point is you can't
add up dollars that you
receive in different periods.
So, in other words,
if I said to you--
if this intertemporal choice
graph was back pizzas and
movies, and I said you have
nine pizza plus movies.
You have a total of nine.
You'd be like, what the
hell does that mean?
It matters if it's nine pizzas
and zero movies or five movies
and four pizzas?
I don't know what that means.
Those are different things.
You can't add them up.
You can't just say
I have nine.
Well consumption over time--
it's the same thing.
You can't just add up your
consumption tomorrow and
consumption today or a dollar
tomorrow and a dollar today.
They're different things.
And so you have to account for
the fact that $1 tomorrow is
worth less than $1 today in
trying to add them up.
And the way we do that is we
actually do it through the
concept of present value.
And the idea of present value
is to translate all future
dollars into today's dollars.
Translate all future dollars
into today's dollars
recognizing the fact they're
less valuable in the future.
So the concept of present value
is the concept of any
future payment's value from
the perspective of today.
And you should know that any
future payment will be worth
less than a payment today.
How much less-- that's what
present value tells you.
How much is a future payment
worth in today's terms?
So suppose that the interest
rate is 10%.
Suppose the interest rate
is 10%, and you want to
have $100 next year.
So you know next year there's
something you want to buy, and
you have to decide how much do
I have to save today to have
$100 in period two.
How much do I have to
save today to have
$100 in period two?
Well if you put in an amount,
PV, into the bank--
you put PV into the bank, then
you know next year you're
going to have-- if you put in
PV in period one, next year
you're going to have
PV times 1 plus i.
You're going to have your PV
plus all the interest you
earned on it, or in our
example, PV times 1.1.
So what that says is that,
basically, you have to put in
100 over 1.1 into the
bank today, or
90.9 dollars, $90.90.
If you put $90.90 in the bank
today, you will have $100
tomorrow or $100 next year--
whenever the periodicity of
the interest rate.
So, basically, more generally,
the present value of any
stream of payments is equal to
that stream's future value--
I'm going to write
it over here.
It's bigger.
The present value of any stream
of payments is that
stream's future value over 1
plus the interest rate to the
t, where t is the year in
which you get the money.
So any future money you get in
year t is worth that amount
you get over 1 plus the interest
rate to the t.
So, basically, the point
is you have to weigt.
Any money you're going to get,
you to weight by how far into
the future it is, just like
if you're adding up these
different goods.
So, essentially, this is kind
of like saying let's add a
converter machine which can
convert pizza into movies.
Then I could say well I'll
just take pizza, put it
through the converter machine,
and that'll tell me how many
movies I have. That's
what a utility
function basically does.
This we're saying the
interest rate is
the converter function.
This present value formula is
the converter function by
which we convert two goods
that are different
into the same good.
You convert them through
this formula.
So, basically, suppose that you
say to me, "Look, loan me
$30, and I'll pay you back $10
a year each of the next three
years." Well I should
say, "Wait a second.
What's the value of that to me?"
Well the present value of
those repayments is I'm going
to have $10 in one year, so
that's $10 over 1 plus i.
Let's say the interest
rate's 10% again.
Next year I got $10.
That's worth 10 over 1 plus 1.
The year after, you're going
to give me 10 more dollars,
but that's worth 10 over 1 plus
1 squared because that's
in two years.
If I had that money today, I
could've invested it, earned
10% and then 10% on that.
And, likewise, the money you
give me in the third year is
worth 10 over 1.1 cubed.
If you'd given me that money
today, I could've invested it
and and earned interest
three times on it.
So the bottom line is
your repayments
are only worth $24.87.
So I've just given you $30 today
in return for a stream
of payments that's only
worth $24.87 today.
I've lost money from that loan
because I gave you the money
today, and you're paying me back
in the future when the
money's worth less.
So, basically, the general
formula we have is that the
present value of any stream of
future payments is that the
amount of the future payment--
let's call it f for any fixed
stream of future payments, $10
forever or $15 forever--
fixed stream of that amount, f,
times 1 over 1 plus i, plus
1 over 1 plus i squared, plus
da da da da, plus 1 over 1
plus i to the t.
So if you're going to pay me a
fixed amount, f, for t years,
here what it's worth
to me today.
You pay me a fixed amount, f,
for t years, it's worth this
much to me today.
I'm accounting for how far
off in the future it is.
Now one important trick we're
going to do now for the rest
of the semester is we're going
to take the trick of saying
this is actually--
well this is a messy formula.
It's actually a rather easy
formula to write down if the
future stream of payments is
infinite, if we have what we
call a perpetuity.
If we have what we call
a perpetuity.
A perpetuity is a future stream
of payments that goes
on forever or long enough that
we'd consider it forever.
Fifty years is probably
good enough.
If you have a perpetuity,
then this formula
can be reduced to--
the present value of any
perpetuity is the amount of
that perpetuity over
the interest rate.
It's just taking the infinite
sum of that product.
They're just taking
the infinite sum.
Those mathematically inclined
will know this already.
But it's just taking the
infinite sum here, you can get
this formula.
So any perpetuity, if you're
getting a payment forever, the
value of that payment today-- so
if I said I'll give you $10
forever and the interest rate's
going to be 10% then
you'd say that's worth
$100 to me.
If I'm going to give you $10
forever at a 10% interest
rate, then you'd say well that's
worth $9.90 the next
year and $8 something the
next year, et cetera.
And if I add all those up, I get
approximately the amount
of the payment over
the interest rate.
So, basically, that is what
determines present value.
Now we can flip this around and
we could say, OK, well, if
that's present value what
determines future value?
Well the future value of getting
a payment today--
So that's the present value of
getting payments tomorrow.
The only other thing is what's
the future value?
What's the future value of
getting a stream of payments
starting today?
So let's say, starting today,
I'm going to get $10.
I'm going to get it for a
certain number of years.
What's that going to be worth
at the end of the day given
that I can save it
along the way.
So, in other words, if you
give me $10 today--
if you give me $10 today, well
then in one year, next year, I
have $11 because I got to save
it at the interest rate.
So if you give me $10 today,
next year I'll have $11.
Now let's say that I
then keep it in the
bank for another year.
Well the next year, it's worth
10 times 1 plus i squared, so
10 times 1.1 squared, 10
times 1.1 squared, or
$12.10 and so on.
So, basically, the point is that
at the end of each year I
earn the interest on my original
$10, plus I earn the
interest on the interest I
earned the previous periods.
So in the long run, at the end
of t years, which of the
future value, is the amount
invested, f, times 1
plus i to the t.
That's your future value.
So if you invest a given amount
of money today for t
years, you end up
with that much.
If you invest a given amount,
f, today you end
up with that much--
And the key point, the key
insight, is the miracle of
compounding.
The miracle of compounding is
the point that you earn
interest on your interest. And
what this means is the earlier
you save, the more you'll
have later on.
So there's an example in the
book which is very important.
So it's actually thinking
about--
let's think for a minute
about retirement.
You might say this
is sort of crazy.
Maybe not crazy for an old guy
like me, but you're thinking
I'm just starting
on my career.
Why would I think about
retirement?
Here's why you want
to think about it.
Let's say, for example, that
you plan to work full time
from age 22 to age 70.
You've got a great idea.
Screw grad school.
You're going right to work.
You've got a great idea,
and you know you want
to retire at 70.
So you plan to work full time
from age 22 to age 70.
And let's say that you want to
save money for your retirement
because you're going
to retire at 70.
You're going to live forever.
You're a healthy young person.
You think you're going to
live forever at 70.
So you want to have money
around when you retire.
And let's say that the
interest rate you
can save at is 7%.
So you can save money for your
retirement at 7%, and that's
your choice.
Now let's consider two different
savings plans.
Savings plan one is that I'm
going to save $3,000.
You're going to save $3,000
right off the bat.
And for the first 15 years
of your working life--
so from 22 to 37--
you're going to save $3,000.
You're going to save $3,000 a
year savings for 15 years, and
then nothing.
And then once you're 37 and
you've got to start worrying
about kids' college and
mortgage, you're not going to
save anything.
So from 22 to 37, when you're
living high on the hog, you've
got no obligations, you're going
to save. Once you're 37,
you've got a house, you've got
kids, you've got things that
are expensive, you're not
going to save anymore.
So then you go to zero.
Zero savings from age
37 to age 70.
That's a pretty bold plan.
You're just going to save 15
years then zero savings.
Well what do you get?
Well after 15 years, if you use
our future value formula,
if you save $3,000 every year,
you work out that you will
have $75,387 in the bank.
So it's more than 45-- is not
just 3,000 times 15 because
you get the compounding.
It's not just 3,000 times 15.
You actually get more than
that because you got the
compounding along the way.
Then if you just let it sit
there in the bank-- you don't
do anything.
No more active savings.
You let it sit there.
Then by the time you retire,
you have that $75,387 times
1.07 to the 33 because you
let that money sit
there for 33 years.
You let that money sit
there for 33 years.
That works out to be $703,010.
So by saving for 15 years-- all
you did was save $3,000
for 15 years, a fraction
of your career.
And you retire with $703,000.
Now we'll contrast that with
an alternative plan.
My alternative plan is I'm going
to save nothing for the
first 15 years because I figure,
like, I'm young.
I'm gonna party.
I'm going to use
the money now.
I'll save later.
I'm nowhere near retirement.
Then I get to 37.
I say, wait a second, I'm
starting to see more mortality.
I better worry about
retirement.
Then I start to save, and I save
for the next 33 years.
So the new plan is zero per year
from age 22 to 37, and
then $3,000 a year
from 37 to 70.
So it's a lot more savings.
You're saving for more than
twice as long, more
than twice as long.
What do you end up with?
You end up with $356,800--
half, just slightly more than
half, than what you end up
with the first plan, even though
you save for twice as
many years.
This is like the parents'
lecture why you should save.
The point is that saving early
lets you ride the wave of
compounding for many,
many years.
Savings late does not
let you do that.
And as a result, you end
up with less money.
I take my kids to the science
museum, and at the science
museum they have these
little ramps.
And you can drop a ball.
And one is flat then steep, and
one is steep then flat.
And the one that's steep then
flat always wins because
there's compounding in
acceleration the same way.
The point is building up early
and then riding that velocity
going forward is a lot better
than starting late.
Questions about that?
So make sure when you get those
jobs, and they offer
you-- we'll talk next time about
savings incentives-- and
they offer you those good 401K
packages, that you take them,
and don't say I'll worry
about that later.
Now, one last thing.
Last thing I want to cover is
that we've ignored, so far,
the whole concept
of inflation.
When I've talked about savings,
I've presumed that
you've saved the money and
it's worth something.
But who the hell knows
what $703,000 will
be worth in 48 years?
What's that even going
to be worth?
How do we even think
about that?
Well we have to account for the
fact that stuff's going to
be more expensive.
So we have to account for
inflation in doing this.
And the way we do this is by
recognizing that what we've
done so far is we've
talked about the
nominal interest rate.
By the nominal interest rate, I
meant the interest rate that
you actually see posted
in the bank.
But what matters, ultimately
for your well-being, is the
real interest rate which is
what your money can do in
terms of actually
buying goods.
So I should not care about how
much money I have next year.
I should care about how many
goods I can buy next year.
The money's just paper.
What matters is what
I can get with it.
That's what matters.
So let's say, for example,
I want to use
all my money on Skittles.
That's just what I want
to use my money on.
So let's say I have $100,
and I want to spend
that money on Skittles.
And let's say Skittles
today are $1 a bag.
And let's say the interest
rate, once again, is 10%.
And let's say there's
no inflation.
Inflation equals zero.
So my choice is I can spend
$100 on Skittles
today and get 100 bags.
So I could have 100
bags today.
Or I can save it, have
$110 tomorrow and
get 110 bags of Skittles.
That's my choice.
That's my trade-off.
Now let's say there's
inflation.
Let's say that prices are rising
at 10% a year, as well.
So the prices are rising
10% a year, as well.
What that means is next year
Skittles cost $1.10 a bag.
So now what's my trade-off?
That means I could
have 100 bags
today or 100 bags tomorrow.
I don't get any more
Skittles tomorrow.
I get 10 more dollars,
but who cares?
Everything costs more.
If everything cost 10% more and
I get a 10% interest rate,
that interest rate is
effectively zero in terms of
what I can buy.
The real interest rate
is the nominal
interest rate minus inflation.
What I care about is
what I can buy.
So I have to take out
of the interest rate
what happens to prices.
Because if prices go up,
it offsets what I'm
earning in the bank.
And so what I care about is I
care about what the bank posts
minus what inflation will be.
So it's even trickier, right?
Because it's not about what
inflation was, it's what
inflation will be.
You'll have to guess what
inflation is going to be.
And so what we care about is
this real interest rate.
And that's why the interest rate
that banks pay, a primary
determinant of it
is inflation.
Right now, we are in the lowest
inflation period this
nation's seen since
World War II.
Core inflation--
we don't really get into
inflation in this course-- but
core inflation, which is
inflation minus some things
which fluctuate a lot, is
basically zero in the US.
So, basically, nominal interest
rates are the same as
real interest rates, and that's
why the interest rates
you see posted are so incredibly
low because there's
no inflation.
In the late 1970's, when
inflation was running at
10-15% a year, interest rates
were 15 to 20% a year.
Now it wasn't that you could
get so much more for your
savings in the 1970's.
It was just that stuff was going
to cost more next year,
so banks, if they wanted to
induce you to save, had to pay
you a higher interest rate.
So, essentially, banks are going
to have to pay you to
get you to put your money in.
If in 1978, when the inflation
rate was 15%--
if banks had offered a 3%
interest rate no one would've
put money in the banks because
you would end up losing
effectively.
Effectively, that's a negative
12% real interest rate.
So what matters is how much
the bank pays you in cash
minus how much more stuff is
going to cost. And that's
often what matters.
Now that's a distinction
we won't spent a lot of
time on later on.
I'll just say interest rate.
I won't say real
versus nominal.
But you've got to know in your
head that what matters is the
interest rate is the real
interest rate, what the bank
pays you, minus how much more
stuff's going to cost.
Questions about that?
Alright, we'll come
back next time.
There is class on Wednesday.
It does matter.
And on Wednesday we're going
to talk about the rest of
capital markets and people's
savings decisions.
