Chapter 8
1
Chapter 10: Investment and the IS Curve
J. Bradford DeLong
- Draft 1.0-1998-04-24: 9,294 words
Demand for investment goods
Demand for investment goods once again
The importance of investment.
As we saw back in chapters 4 and 5 level of investment spending is one of the
principal determinants of long-run economic growth: the rate of savings and
investment plays a powerful role in determining the steady-state capital-output
ratio, and the steady-state capital-output ratio determines the position of the
steady-state growth path.
But here we focus on a different role for investment. Changes in investment
spending are the principal force driving the business cycle. Investment is one of
the most volatile components of GDP. Reductions in investment have played a
powerful role in every single recession, and increases in investment have paid a
powerful role in every single boom, in the United States for as long back in time
as we know.
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2
[Figure: Gross and net investment as a share of GDP]
Net Investment as a Share of National
Product
Percent
12%
Inves tment high because
of extremely low inflationadjusted interest rates
Reces sion of 1991-92;
inves tment low because
of depres sed animal
spirits
Reagan-era
optimism
10%
8%
6%
Exces sive optimis m
of the 1960s
4%
2%
Inves tment low because of
extremely high inflationadjusted interest rates
0%
196 0
197 0
198 0
199 0
Lowered interest
rates drive
inves tment
recovery
Chapter 8
3
Net and Gross Investment as Shares of
National Product
20 %
Gross Investment
15 %
10 %
Net Investment
5%
0%
19 60
19 70
19 80
19 90
Fluctuations in investment have two sources. Some fluctuations in investment
are triggered by changes in interest rates. Lower interest rates mean higher
investment spending.
Other fluctuations in investment are the result of investors' expectations about
future economic growth, future level of profits, future levels of risk, and their
willingness to gamble or their desire to avoid risk.
There are three kinds of investment. The first is the purchase and installation of
new business machinery and equipment; the second is the construction and
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4
purchase of a new building (or the repair of an old one); and the third is a change
in the level of business inventories.
However, the differences between different forms of investment are secondorder: even though the motives that lead businesses and builders to engage in
the three different kinds of investment are somewhat different, they all depend
on expectations of future economic activity and on the level of interest rates.
Little is lost if we take one kind of investment to be the canonical--representative-case. So economists usually discuss all investment as if it were undertaken a
business trying to decide whether, and how much, it should spend expanding its
capital stock to make itself more productive and more profitable.
[Box: Details: What is investment?
Recall that when economists use the term "investment," they mean
something special: they mean transactions that add to the capital stock
of the economy as a whole, and increase the economy's potential
output. To an economist, investments are (a) the purchase and
installation of new business machinery and equipment, (b) the
construction and purchase of a new building (or the repair of an old
one), and (c) a change in business inventories.
There are some differences between the motives that lead to different
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kinds of investment. Home builders look further out into the future in
deciding what to build than corporation managers do in deciding what
capital goods to buy. Decisions to increase inventory levels can be
reversed quickly, at low cost. But many forms of fixed investment are
irreversible--hence are likely to dry up when risk is great, or when
people believe that new information is likely to arrive quickly.
However, these differences between different forms of investment are
second-order. Little is lost if we take the canonical case of investment
to be a business trying to decide whether, and how much, it should
spend expanding its capital stock to make itself more productive and
more profitable.
Interest rates
Whenever we think about investment we have to think about interest rates.
Undertaking an investment project is a business's way of trying to preserve its
capital and make it grow. The alternative to undertaking the investment project
that is open to the business is for it to invest its money, instead, in the financial
markets--where (if invested in bonds) it will earn the market interest rate or (if
invested in stocks) it will earn the market interest rate, plus the extra equity
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premium for taking on the risk that the company might lose money, plus or minus
whatever extra capital gain or loss occurs as other investors change their
opinions about the future and thus what they are willing to pay for stocks.
Thus the opportunity cost of undertaking an investment project is intimately tied
up with the interest rate.
The Idea of Present Value
Is it worthwhile making a $10 million investment today that will
return an operating profit of $13 million in five years? Present
value is a way of figuring this out by asking the question "what else
could you do with the money?"
Invest
$10
$13
Loan it
out in
the bond
market
4% interest rate
6% interest rate
8% interest rate
Chapter 8
7
Long and short-term interest rates.
But which interest rate? There are many.
Investment goods are durable goods: they almost always last for more than five
years, and they can last for a long time indeed. Therefore whenever a
corporation's financing committee considers whether to undertake an
investment, it must compare the potential profits to the opportunity to make
money from a long-term commitment of the funds elsewhere.
Thus the opportunity cost of undertaking an investment is not a short-term
interest rate--not the interest rate paid on a three-month or a six-month loan--but
a long-term interest rate: the interest rate on a long-term loan for a period of a
full decade or more.
Long and short-term interest rates are different. They do not move in step
together. Usually the long-term interest rate is higher than the short-term interest
rate, but not always.
[Figure: short and long-term interest rates in the U.S.]
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8
Real and nominal interest rates.
Moreover, the interest rate that is relevant to the opportunity cost of
undertaking business investment is a real, not a nominal interest rate.
Suppose, for example, that the prevailing nominal interest rate is 5.9% and
inflation is 3% per year. If you borrow $1,000 for one year this year, you will have
to repay $1,059 next year. But the 3% inflation means that products you can sell
for $1,000 today will fetch $1,030 next year. So next year you would only have to
scrape up an an extra $29 dollars--2.9% of your loan--to repay your lender. That
2.9% is your real interest rate.
The nominal interest rate tells us how many dollars we must repay in the future
if we borrow in order to have one more dollar today. But inflation changes a
dollar's purchasing power over time. So, when calculating how expensive it is to
borrow, what we really want to know is how much power to purchase future
goods and services we must give up in order to get more power to purchase
goods and services today. That is what the real interest rate tells us.
If inflation is high, then paying high interest rates in the future is not a bad deal.
Why? Because we would only need to sell very few goods to get the dollar bills
to repay the loan.
Because the prices (and operating profits) of a business are likely to increase as
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9
the price level increases, a business will not be more averse to investing when the
interest rate is 10% and inflation is 7% per year than when the interest rate is 5%
and inflation is 3% per year. It is the nominal interest rate minus the expected
inflation rate that is relevant to investment decisions, not the nominal interest
rate all by itself.
[Figure: real and nominal interest rates]
[Box: Examples] Calculating real interest rates.
[To be written]
Safe and risky interest rates.
Lending money carries an element of risk. Perhaps the borrower will go
bankrupt before the loan is due. Perhaps the creditor will find themselves last, or
nearly last, in line as a small amount of left-over assets are divided up.
Thus financial institutions loaning money are keenly interested in the financial
health of those to whom they lend. And the riskier they believe the loan is--the
larger the possibility of a bankruptcy or a debt rescheduling appears to be--the
higher is the interest rate that lenders will demand to compensate them for risk.
Hence the interest rate that a firm faces when figuring out whether a particular
investment project promises returns in excess of opportunity cost is the interest
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10
rate charged to risky borrowers, not the interest rate charged to safe borrowers
(like the U.S. government) to whom people lend when they want to sleep easily
at night.
The premium that lenders charge for loans to companies rather than to safe
government borrowers is called the risk premium. The risk premium is a function
of the perceived riskiness of businesses in the economy. Financial and economic
disturbances can cause large and swift moves in the risk premium. Recall the
sharp increase in the risk premium in August of 1998, when the Russian
government postponed payment on some of the loans it owed.
[Figure: bond riskiness interest spreads]
[Box: Examples] From short-term safe to long-term real interest rates
[To be written: Japanese interest rates in the 1990s]
Interest rates and bond prices.
In almost all of this book we focus on real interest rates--interest rates adjusted
for the rate of inflation, and symbolized by the letter r. But in this subsection we
focus on nominal interest rates--interest rates in terms of money, symbolized by
the letter i. We do this because the bonds and loans that are traded and made in
the real world are nominal securities: they are promises by the borrower or debtor
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11
to pay back the lender or creditor fixed sums of dollar bills at times in the future;
they are not promies to pay back fixed amounts of inflation-adjusted purchasing
power.
Thus calculating the relationship between interest rates and bond prices is much
easier if we work in nominal terms, and so we focus in this section on nominal
interest rates.
Of course, once you have calculated the relationship between interest rates and
bond prices and then want to look outside at the effect of these variables on the
rest of the economy, you will immediately want to subtract the inflation rate
from the nominal interest rate, and so return to focusing on real interest rates.
Whenever you go to the bank--for a student loan to pay for college, for a car loan,
or a mortgage, or (heaven forbid!) to consolidate your credit-card debt--you take
out a loan. You tell the bank how much you want to borrow, and for what period.
The bank tells you what interest rate you will have to pay.
For example, if you borrow $10,000 for a term of one year at an interest rate of
7%, then a year from now--when your loan matures--you will have to pay back
the bank the $10,000 "principal" sum that you borrowed, plus 7% x $10,000
equals $700 in "interest": a total of $10,700. As interest rates change, the interest
sum that the bank will charge for a given principal amount and a given term will
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12
change too. (And the terms on which the bank offers to accept deposits--the
interest that it offers you will change as well.)
But in our economy much of the money that is borrowed by governments and
corporations is borrowed not through loans but through bonds. When the interest
rate changes, the interest payment attached to a loan changes as well, while the
principal amount remains fixed. But when the interest rate changes, both the
interest payments--the so-called coupon--paid during the bond's lifespan and the
principal payment made when the bond matures by the issuer of a bond to the
holder of the bond remain fixed: what changes is the market price of the bond.
How does this work? Consider the simplest of bonds, a so-called discount bond-in this case, a $1,000 one-year Treasury bill (or T-bill) issued by the U.S. Treasury.
A $1,000 one-year T-bill is a promise by the U.S. Treasury to pay the holder of the
bond $1,000 on a date certain one year from the date on which the bond is issued.
What will be the market price of a newly-issued T-bill?
Suppose that the investors thinking about issuing the bond could also loan their
money out directly, to a bank or a corporation, at a nominal interest rate of i%.
Then for each one dollar they invest in such a loan, they receive 1+i% after one
year. How much would investors receive for each dollar that they invest in
buying a bond?
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13
Suppose that the newly-issued T-bill sells at a price of $P. Then after a year the
government pays the holder of the T-bill $1,000. So that for each dollar that
investors commit to buying T-bills, they receive $1,000/$P after one year.
Then the price of the T-bill will be whatever value makes the returns to investors
from making loans and buying bonds equal. The price of the T-bill will make the
equation:
1 + i% = $1,000/$P
true. And so:
$P = $1,000/(1+i%)
Why? Because we see both loans being made and bonds being bought. If the
price of bonds was lower than $1,000/(1+i%), then investors would have an
opportunity to make money by using all their wealth to buy bonds--they would
even borrow money from banks at the prevailing interest rate i% and use that
money to buy T-bills. If the price of bonds was higher than $1,000/(1+i%), then
the same process would work in reverse: the government would find nobody to
buy its bonds, because they would all be making loans to well-capitalized banks
and sound corporations at the interest rate i% and so earning more than they
could by buying bonds.
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14
This is called an arbitrage argument--that a market price must have a particular
value, because we see both loans being made and bonds being bought--and if the
market price were not at that particular value, then investors would have to be
mad to make loans (if bond prices are too low) or mad to buy bonds (if bond
prices are too high).
The fact that the price of bonds and the prevailing interest rate are related by:
$P 
$1,000
1 i%
means that whenever interest rates go up, bond prices fall--and vice-versa: when
interest rates fall, bond prices rise. This makes it hard to watch or listen to reports
about the bond market. Does someone saying "bonds fell" mean that bond prices
fell (and hence interest rates rose), or do they mean that interest rates fell (and
hence bond prices rose)? (Usually they mean the first.)
A short-term T-bill is at one extreme of the spectrum of possible bonds that a
company or a government might issue. At the other extreme is a kind of bond
called a consol (short for "consolidated obligations" because these bonds were
originally issued by the British government in the eighteenth century when it
consolidated its debt after wars). A consol is a bond that never matures: there is
no date certain in the future at which the bond contract expires, and at which the
Chapter 8
15
issuer pays back the principal to the holder. Instead, a consol is a promise by the
issuer to each year pay a fixed nominal amount to the holder year after year
forever.
What is the price of a consol that pays $C (for coupon) each year? We can
determine it by making the same kind of arbitrage argument. Suppose that the
long-term nominal interest rate is il% (long-term because the consol is a longterm security, and so the short-term interest rate is not very relevant. And let us
consider a bond trader who wants to figure out which is better: to lend out $1 at
the prevailing long-term interest rate il% and to keep relending the money to
another borrower whenever one borrower pays back the loan, or to buy a consol
selling at a price $Pc and to collect the coupon payment $C--"clip the coupons".
In the first case, the bond trader receives il% in interest each year for each dollar
that he or she commits to long-term loans. In the second case the bond trader
commits $Pc and receives $C in each year, so he or she receives $C/$Pc in interest
each year for each dollar committed to buying consols. If the first rate of return is
higher, then everyone will lend out their money and no one will buy consols. If
the second rate of return is higher, then everyone will buy consols and no one
will lend out their money. Since we see both happening--both long-term loans
being made and long-term bonds being bought and held--the two must be equal:
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16
$C
c
$P
i%
l
And we can turn this equation around into:
$P 
c
$C
i%
The price of a consol is equal to the consol's coupon--interest--payment divided
by the long-term nominal interest rate.
You can see that the price of a long-term bond--like a consol--is much more
sensitive to fluctuations in the nominal interest rate than is the price of a shortterm bond--like a T-bill. If the short-term nominal interest rate is 5% per year and
then rises to 6% per year, the price of a one-year $1,000 T-bill falls from:
$P 
$1,000 $1,000 $1,000


 $952.38
1 i% 1  5%
1.05
$P 
$1,000 $1,000 $1,000


 $943.40
1 i% 1  5%
1.05
to:
a loss of $9 on an approximately $1,000 investment from a 1 percentage point rise
in nominal short-term interest rates. By contrast, if the long-term nominal
interest rate is 5% per year and then rises to 6% per year, the price of a five
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17
percent consol--a consol that pays out $50 per year as its coupon payment--falls
from:
$P 
$50 $50 $50


 $1,000
l
i % 5% 0.05
$P 
$50 $50 $50


 $833.33
l
i % 6% 0.06
c
to:
c
a loss of nearly $167 on a $1,000 investment from a 1 percentage pint rise in
nominal long-term rates. The price of a long-term bond like a consol is thus
ninety times more sensitive to an equal percentage point shift in interest rates
than a short-term bond like a one-year T-bill.
It is true that an average jump in nominal long-term rates is only one-third as
large as an average jump in nominal short-term rates. Even so, a typical upward
shift in nominal interest rates will push down the price of a consol thirty times as
much as the price of a T-bill. Long-term bonds are risky securities in which you
can lose your money. When interest rates go up, long-term bond prices go down-a lot.
The inverse relationship between interest rates and bond prices means that
whenever interest rates go up, bond prices fall--and vice-versa: when interest
Chapter 8
18
rates fall, bond prices rise. This makes it hard to watch or listen to reports about
the bond market. Does someone saying "bonds fell" mean that bond prices fell
(and hence interest rates rose), or do they mean that interest rates fell (and hence
bond prices rose)? (Usually they mean the first.)
[Box: Details] Talking like a bond trader
[To be written]
[Box: Examples] Calculating changes in interest rates and movements in bond prices
[To be written]
[Box: Details] Discount and coupon bonds
[To be written]
[Box: Details] Interest spreads: the difference between what banks pay and what banks
receive.
[To be written]
Chapter 8
19
The yield curve.
Graph the pattern on any one day of how interest rates vary with how long the
maturity of the loan or the bond is. Plot the interest rate on the vertical axis, and
the length before maturity on the horizontal axis. That pattern, for any one day,
is that day's yield curve.
Bond Yield Curves
Yield to Maturity
8%
3-Mo
Bill
7%
Mar-92
6-Mo
Bill
Dec-96
6%
5%
30-Yr
Bond
10-Yr
Note
3-Yr
Note
4%
0
2
4
6
8
Duration of Bond (Years )
10
12
14
Usually the yield curve slopes upward: usually long-term interest rates are
higher than short term interest rates, because bonds and loans with a long
maturity are perceived to have higher risk. In late 1992 the yield curve was very
steep: long term loans carried much, much higher interest rates than short-term
loans. The premium in the interest rate that the market charges on long-term
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20
loans vis-à-vis short term loans is called the term premium.
term premium  r  r
s
Where r is the long-term, and rs is the short-term real interest rate.
The term premium is closely related to what financial market speculators and
traders expect to happen to short-term interest rates in the future. When they
expect short-term interest rates to rise steeply, the term premium is large. When
they expect short-term interest rates to rise slightly in the future, the term
premium is small.
When they expect short-term interest rates to fall, the term premium is negative.
Short-term interest rates are higher than long-term interest rates. Financiers call
this an inverted term structure--because more often than not long-term rates are a
little above short term rates.
[Box: Details] What determines the term premium
What determines the value of this term premium? Consider a simple
two-period model in which the periods are "now" and the "future".
Bankers make long-term loans that fall due in two periods. Bond
traders buy and sell long-term bonds that fall due in two periods. The
real interest rate paid on these long-term loans and bonds is the long-
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21
term real interest rate r. Bankers also make short-term loans (and bond
traders also buy and sell short-term bonds) that mature in just one
period.
Someone thinking about buying a long-term bond (or making a long
term loan) knows that for each real dollar they invest in such financial
instruments today, they will after two periods have:
gross return 1  r  r
Each period they will receive the long-term real interest rate on their
investment, r.
Someone thinking about buying a short-term bond today (or making a
short-term loan) knows that for each real dollar they invest in such
financial instruments today, they will after the end of the first period
(the one that is going on "now") have 1 + rsn: "r" for the real interest
rate, "s" because it is the rate paid on a short-term loan, and "n"
because it is the interest rate paid in the "now" period. But their capital
will then, at the start of the second period (the one that will happen in
the "future") be lying idle. The natural thing to do then will be to invest
it again in another short-term bond (or make another short term loan-this time at the short-term interest rate that will prevail in the future,
Chapter 8
22
rsf. So after two periods someone who chooses today to invest their
money in short-term securities will have:
gross return  1  r n  r
s
s
f
For each real dollar that they invested at the start of the first period.
What will a flint-eyed money-maximizing rational bond trader do?
The first complication is that he or she doesn't know today what the
future short-term real interest rate rsf will be when the time to reinvest
the principal arrives. The best he or she can do is form an expectation
now--En--of what the future short-term real interest rate will be: En{ rsf}.
Thus the flint-eyed money-maximizing rational bond trader has to
decide whether to invest for the long-term or to invest for the shortterm (and then, later, to reinvest). The returns from investing for the
long-term will be greater if:
long term gross return  1  2r  1  r sn  En r s f  short term return
Or, defining En{r}, the expected change in the short-term real interest
rate, as the difference between expected future short rates En{ rsf} and
current short rates rsf:
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23
Expected change in short rates  Enr
s
 E r  r
s
n
f
s
n
The returns from investing for the long-term will be greater if:
r  rn 
s
En rs 
2
And they will probably decide to invest for the long-term. If:
r r 
s
n
En rs 
2
Then the returns from investing for the short-term will be greater, and
they will probably decide to invest for the short-term.
In equilibrium there are both short-term and long-term bonds held, and
short-term and long-term loans made. So in equilibrium the typical
bond trader and bank loan officer must think that the expected returns
from long-term and short-term financial investments are roughly
equal. In equilibrium:
r r 
s
n
En r s
2
The term premium r-rs is equal to the expected change in short-term
interest rates over the life span of the loan, weighted by the proportion
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24
of the loan's time span over which the changed short-term interest rate
will apply.
In other words, the term premium tells you how bond traders expect shortterm interest rates to move in the future. If financiers are buying 2-year
bonds at, say, 5.75%--when they could instead buy 3-month T-bills
every quarter for two years--then they must believe that either
portfolio strategy will average out to about 5.75% over two years, or
else they would all be crowding into one security or the other. Demand
for the one would rise, demand for the other would fall. And the
interest rates on them and on loans of that duration would change
until once more it looked to bond traders that the two strategies were
equally attractive.
Similarly, if bond traders are buying 3-month T-bills at, say, 4% when
they could instead buy 2-year bonds at 5.75%, then they must expect
that higher short-term rates a year and a half in the future--say, 7.5%-will balance out today's low rates to average out to 5.75%.
This is the expectations theory of the term structure: the long-term interest
rate is the average of what bond traders expect future short-term rates
to be for the duration of the long-term loan. The term premium tells us
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25
how much bond traders are expecting the average short-term interest
rate to rise (or fall) over the duration of the long-term loan.
[Box: Examples] Calculating interest rates using the expectations hypothesis of the term
structure
[To be written]
[Box: Policy] The term structure and interest rate expectations at the end of 1992
[To be written]
[Box: Details] The average slope of the term structure
Even when short-term interest rates are expected to be constant, the
yield curve will probably slope upwards. The expectations theory of
the term structure is a good approximation, but it is not totally
accurate because long-term bonds are slightly riskier than short-term
bonds. [To be written]
Chapter 8
26
Present value, interest rates, and investment
Present value.
Perhaps the most useful tool economists have to analyze how interest rates affect
investment decisions is the concept of present discounted value. A business
manager considering whether he or she should buy a new machine or construct a
new factory should compare the returns from the investment project to the
opportunity cost of the purchasing power that needs to be used to buy the
machine or build the factory. And the best and easiest way to calculate whether
the returns on the purchasing power committed exceed the opportunity cost is to
calculate the present value of the future extra profits from the investment project.
If the present value of the expected future profits exceeds the cost today of
undertaking the project, then the manager should go ahead. If not, then not.
But we do not observe present values directly. Instead, we must calculate them
from information about future profits and interest rates.
Making present value calculations.
If the one-year nominal interest rate is i%, then one dollar committed to making
loans today or buying bonds today will yield a total of 1+i% dollars in one year.
Thus we can say that $1 today is worth $(1+i%) in one year. They are equivalents.
They have the same value.
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27
We can turn the argument around. Suppose that we are interested in figuring out
what amount $x we would have to lend out today (or use to buy bonds today) in
order to have $1 in a year. The answer is that x is equal to 1/(1+i%):
$x 
$1
1  i%
In this sense, $1 next year is worth $1/(1+i%) dollars this year. They are
equivalents. They have the same value.
So we say that $1/(1+i%) today is the present value of $1 one year in the future.
The reason is obvious: the first sum is in the present, and it has the same value as
$1 a year from now--in the sense that a flinty-eyed rational utility-maximizing
individual able to borrow and lend at the prevailing interest rate of i% doesn't
care which he or she gets when offered a choice between $1 next year and
$1/(1+i%) today.
You will also hear economists talk about "expected present value"--because we
do not know the future with certainty, and all of our calculations are based on
our expectations of it. You will also hear economists talk about "discounted
present value" because the factor 1/(1+i%) that we use in calculating present
values is less than one, and thus sums to be paid in the future sell at a discount to
their face value today.
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28
Suppose, however, that you want to calculate the present value of something, but
that the something isn't a simple one-time payment of $y next year. Suppose that
you are the manager of a business, and that your staff have brought to you plan
to invest $10 million expanding your factory, and that they project that if this
expansion is started now it can be finished in time for this year's busy season and
it will raise annual profits by $500,000. Furthermore, because the market is
growing, they project that the real inflation-adjusted profits from the expansion
will then rise by 3% per year as far in the future as they can see.
What is the expected present value of the future profits to be made from
undertaking this investment? Let's consider a slightly more abstract problem: the
real profits are $C, and the real profits are projected to grow at a rate of g% per
year as far in the future as they can see.
Take, first, the special case in which the growth rate of profits g% is zero. Then if
you look at the stream of profits from the factory--a constant stream of $C per
year--you will all of a sudden recognize that undertaking this expansion looks a
lot like buying a consol: a consol also paid you a constant stream of $C per year.
And we know that the equilibrium price of a consol is just the annual interest
payment divided by the interest rate. So the present value of this--constant,
perpetual--stream of profits of $C per year is just equal to $C divided by the
Chapter 8
29
interest rate.
But which interest rate? It must be the long-term interest rate because this factory
expansion is a long-term project. It must be the real interest rate because the
profits result from the sale of real commodities (and if inflation raises the price
level, it will raise the nominal dollar sum of profits by the same proportion). And
it must be the risky interest rate appropriate for valuing risky investments-because the profits from expanding a factory are a lot more variable and volatile
than are the coupons to be clipped from loaning money to a stable, secure
government.
So when the expected growth rate g% of real profits is zero, then the present
value of the profits from the factory expansion is, with r% standing for the longterm, real, risky interest rate:
$PV 
$C
r%
When the expected growth rate is not zero, the present value of the profits is very
similar: you divide this year's profits not by the appropriate long-term riskyinvestments real interest rate, but by the difference between the real interest rate
and the growth rate:
Chapter 8
30
$PV 
$C
r%  g%
And once you have the present value, then deciding on whether to make the
investment is easy: you make the investment if the present value of future profits
is greater than the cost today of the project. If the present value is greater, then
the investment project returns value in excess of its opportunity cost. If not, not.
In the case of our example above, the cost was $10 million, the increase in annual
profits $C was $500,000, and the real growth rate of profits g% was 3% per year.
Thus if the appropriate real interest rate is 5% per year, then the present value is:
$PV 
$C
$500,000 $500, 000


 $25,000,000
r%  g% 5%  3%
2%
The present value of future profits is far in excess of the current cost, so the
investment definitely should be undertaken. But if the appropriate real interest
rate is 8% per year, then:
$PV 
$C
$500,000 $500, 000


 $10,000,000
r%  g% 8%  3%
5%
Then the project is on the knife-edge: there is no gain from either undertaking it
or not undertaking it. And if the appropriate real interest rate is 10% per year,
then:
Chapter 8
31
$PV 
$C
$500,000 $500,000


 $7,142,857
r%  g% 10%  3%
7%
And the investment project is definitely not worth making: future profits are not
high enough.
The values of investment projects look a lot like the values of bonds: in both
cases, the higher the interest rate, the lower the value today of the payments that
the holder or investor will receive in the future.
[Box: Details] Present values when real profits are growing at g% per year
[To be written]
Investment and present value.
A profit-making business invests in the hope of making more money in the
future. A business will seek to make only those investments that are profitable. A
business that does not think it can make money by investing in new plant and
equipment will not do so. It can do other, more productive things with its
resources.
Consider a firm that has a number of possible investment projects it might
undertake:
Chapter 8
32
(A) Spend $10 million improving its distribution system, increasing profits by
$400,000 a year.
(B ) Spend $30 million building a new factory, increasing profits by $2 million a
year.
(C) Spend $5 million upgrading machine tools, increasing profits by $1 million a
year.
The firm's managers will ask whether such expenditures promise profits in
excess of the financial resources' opportunity cost, and so have a present value in
excess of their current cost.
Suppose that the long-term real interest rate at which the firm could borrow
money (or could lend its surplus cash) is 3 percent. Then the firm's investment
committee would think as follows.
"Project A… produces $400,000 a year in profits… it would cost us $300,000 a
year in interest payments to borrow the $10 million to finance it… that is a
$100,000 a year increase in net profits…
"Project B… produces $2 million a year in profits… it would cost us $900,000 in
interest to finance it… that is a $1.1 million a year increase in net profits…
"Project C… produces $1 million a year in profits… it would cost us $150,000 a
Chapter 8
33
year to finance it… that is a $850,000 a year increase in net profits…
"We should do them all."
When the long-term real interest rate is at 3%, the firm will want to undertake all
three investment projects--and will spend $45 million this year on investment.
But suppose that the long-term real interest rate is 7%, or 12%?
At a long term interest rate of 7% the annual interest cost of financing project A is
$700,000, so A would reduce net profits by $300,000 a year. Similarly project B
would cost $2.1 million a year in annual interest payments, and so would reduce
net profits by $100,00 a year. Only project C is still profitable at a 7% interest rate.
So the firm will undertake only $5 million this year in investment.
And at an interest rate of 12% the firm will undertake no investment at all.
[Figure: Firm investment as a function of the interest rate]
Add up all the investment decisions made in all the conference rooms of all the
firms across the economy, and find that investment spending is a decreasing
function of the long-term real interest rate:
I  I(r )
l
And when we want to simplify our lives we do so by assuming that the
Chapter 8
34
investment function is simple and linear:
I  I0    r
l
Where I0 is the amount of investment that would be undertaken if the interest
rate were very very low, and  is the amount by which each unit increase in the
long-term real interest rate shrinks investment spending.
[Figure: Investment as a decreasing function of the long-term real interest rate]
Investment and the stock market.
Interest rates, profits, present values, and the stock market
The stock market provides a convenient and easily-observed summary measure
of how businesses' and investors' expectations and interest rates combine. It is
the best single measure of how interest rates and expected profits together put
upward or downward pressure on the level of investment.
Chapter 8
35
Suppose it costs $10 million and 5 years to build a
factory that would be worth $12 million today...
At a nominal interest
rate of 7% and an
inflation rate of 2%
Borrow $10 million for five years,
and in five years you owe:
$
(14,025,517)
But five years of inflation have
raised the dollar value of the factory
to:
$
13,248,970
Net Profit
$
(776,548)
At a nominal
interest rate of 7%
and an inflation
rate of 7%
$ (14,025,517)
$
16,830,621
$
2,805,103
When real interest rates are low, the stock market is high. Because other options,
such as bonds, are less attractive, investors are eager to hold stocks, and they bid
the prices of stocks up. When real interest rates are high, the value of the stock
market is low. Because investors can buy bonds that offer relatively high (and
investors hope safer) returns, stocks appear less attractive--and so investors are
not willing to pay as high prices for them.
The stock market is high also when the future looks bright--when today's profits
are high and when the likelihood that profits will grow rapidly in the future is
high, the stock market will be high. When the future looks gloomy, the stock
market will be relatively low.
A simple shorthand equation to remember for calculating what the value of a
stock market index (or of an individual stock) will be is:
Chapter 8
36
P
D
r  g
Where D stands for the annual dividends paid by the corporation on a share of its
stock, P stands for the price of a share of stock, g is the expected annual rate of
growth of dividends, r is the long-term real interest rate on bonds, and  is the
risk premium--an extra expected return that investments in a stock market index
like the Standard and Poor's composite index or that investments in an
individual stock require in order to make investors happy holding such a
potentially risky security.
The risk premium  varies from stock to stock, from stock market index to stock
market index, and from decade to decade--and economists' theories are not very
good at accounting for the level of this equity risk premium, or of changes in it.
The stock market and investment.
A high level of the stock market is not very effective as a cause of higher
investment. Corporations raise relatively little money for new investments by
issuing new shares of stock. Instead, the funding for new investments comes
primarily from sources internal to the corporation--depreciation allowances and
Chapter 8
37
retained earnings--and secondarily from borrowings from banks and the issue of
bonds.
However, things that make the stock market high--optimistic expectations of the
future and low real interest rates--are also the things that make the net present
value of investment projects positive. Think of the stock market as a thermometer
of the economic forces determining investment. When the reading on the
thermometer is high, the water is hot. But a high reading on the thermometer is
not the cause of the water's hotness.
[Box: Details] The investment accelerator.
A second reason for a firm to invest is if it has the money to spend. For
a firm to raise money from outside itself to finance investment-whether by issuing bonds or stocks or by borrowing from banks--is
quite expensive. Lenders or stock purchasers may fear that they are
getting themselves into a situation that they do not fully understand.
Also, a firm must bear the costs of undertaking the transaction.
As a result of both factors, many businesses prefer to wait to invest
until their own cash flow from their own lines of business will cover
the costs. There are investment projects that do not have a positive net
Chapter 8
38
present value when future profits are evaluated at the interest rate that
corresponds to the cost of obtaining funds by borrowing from outside,
but that do have a positive net present value when future expected
profits are evaluated at the internal cost of funds.
Because investment depends on firm cash flow, businesses tend to
invest more in goods times than in bad. This is a factor that tends to
raise the economy-wide marginal propensity to spend above the
marginal propensity to consume (of course there are other, more
powerful factors (the propensity to import that drives a wedge
between total spending and spending on home-produced goods and
services, and the tax rate that drives a wedge beween total incomes
and disposable incomes) that push the economy-wide marginal
propensity to spend below the marginal propensity to consume.
Investment and aggregate demand
From the interest rate to investment to aggregate demand.
Why we use the IS curve.
The IS curve is a tool that macroeconomics courses use in analyzing booms,
recessions, inflation, and unemployment. It got its name from the first
Chapter 8
39
economists to analyze it, who thought of it as an "Investment-Savings" curve. The
equilibrium level of aggregate demand and GDP--the level that the incomeexpenditure diagram exists to calculate--depends on investment spending. And
investment spending depends on the interest rate. So each possible level of the
interest rate is associated with it a different GDP level. That relationship is the IS
curve.
This diagram has been the workhorse economists have used to understand
booms and recessions: fluctuations in national product, unemployment, and
interest rates. The IS curve shows how equilibrium aggregate demand and
national product vary with the interest rate. The Federal Reserve chooses a shortterm interest rate. Expectations working through the term structure and the stock
market generate the long-term real interest rate that tells us where on the IS
curve the economy is.
Thus we can figure out the level of aggregate demand and the level of the
interest rate that together match planned expenditure to national product given
interest rates.
Chapter 8
40
From the income-expenditure diagram…
The income-expenditure diagram shows how (for a fixed level of the interest
rate--thus of investment--net exports, and government purchases) the
consumption function determines aggregate demand as a function of national
product. It tells us (for that interest rate) equilibrium national product: the level
Chapter 8
41
that is equal to aggregate demand. Each value for the interest rate has a different
such diagram associated with it. The IS curve captures the information in all of
them.
The IS diagram plots national product (or national income, or output: here the
terms are synonymous) along the horizontal axis. It plots the interest rate on the
vertical axis. The income-expenditure diagram plots national product on its
horizontal and total expenditure--aggregate demand--on its vertical axis.
…To the IS curve.
How do you get from the income-expenditure diagram to the IS curve? Start
with the interest rate, determine investment, plug that level of investment into
the income-expenditure diagram, and calculate equilibrium national product.
Then the interest rate you started with and the national product level you ended
with are one single point on the IS curve.
Repeat the process, for as many different possible interest rates as you wish. Plot
the points on the IS diagram. And then connect them. You have your IS curve.
Chapter 8
42
Note that this curve you have just plotted--this IS curve--slopes downward as
you move to the right along the graph. The higher the interest rate, the fewer the
investment projects that have a positive net present value, and thus the lower is
investment spending. But lower investment spending means that for any given
level of national product, aggregate demand is lower on the income-expenditure
diagram. And so the equilibrium level of output is lower as well. Thus the IS
curve slopes downward: A higher interest rate on its vertical axis reduces the
Chapter 8
43
equilibrium level of output on its horizontal axis.
Determining the position and slope of the IS curve
The slope of the IS curve.
What is the downward slope of the IS curve? That depends on three things: (a)
How much does an increase in the interest rate reduce investment spending? (b)
How large is the multiplier? (c) How do changes in international trade driven by
changes in interest rates affect total demand.
Consider the first two factors. The larger is the multiplier, the larger is the total
impact on aggregate demand set in motion by a given change in investment
spending. Lowering interest rates raises investment, which raises demand, which
raises production, which raises employment, which raises consumption
spending, which further raises demand. We spent all that time in chapter 7
deriving the multiplier because the multiplier tells us how much a one dollar
increase in government purchases, exports, or--in this case--investment
ultimately raises the equilibrium level of aggregate demand.
But the slope of the IS curve--how much a given change in interest rates changes
the equilibrium level of production and aggregate demand--depends on more
than how large a change in the equilibrium level of production is generated by a
Chapter 8
44
given change in investment. It depends on how large a change in investment is
generated by any particular change in interest rates. You need to multiply these
two factors together to calculate that portion of the slope of the IS curve
determined through the effect of lower interest rates on investment.
Net exports and the slope of the IS curve.
But there is a third factor. Changes in the interest rate change currency
speculators' demands as well, and lead to an appreciated real exchange rate (if
the domestic interest rate rises) or a depreciated one (if the domestic interest rate
falls). An appreciated real exchange rate means lower net exports--and lower
aggregate demand. A depreciated real exchange rate means higher net exports-and higher aggregate demand. Thus changes in interest rates generate changes in
net exports that reinforce the already-covered effects of changes in investment on
equilibrium aggregate demand. A change in interest rates has a bigger effect on
the equilibrium level of output than one would calculate from the effect of
interest rates on investment alone.
[Figure: IS curve and its slope]
Chapter 8
45
The slope in algebra.
More quantitative precision can be gained by repeating the same discussion-although in algebra, not in words. The slope of the IS curve depends on (i) how
much of a change in aggregate demand is generated by a change in investment,
(ii) how much of a change in investment is generated by a change in interest
rates, and (iii) how much of a change in aggregate demand is generated by the
effects on the exchange rate and on net exports of the shift in interest rates.
So multiply the responsiveness of investment to a change in the short-term real
interest rate--the parameter  in the investment equation:
I  I(r)  I0  r
by the effect of a change in autonomous spending on the equilibrium level of
aggregate demand--the factor 1/(1-c*) = 1/(1-c+tc+) in the equilibrium incomeexpenditure equation:
Y 
A
A

1  c * 1  c  tc  
And in their product:

1


1 c * 1 c  tc  
you have the impact of these first two factors on the slope of the IS curve--the
Chapter 8
46
change in equilibrium output generated by a change in the interest rate acting
through its effect on the level of investment:
Y 
  r


r
1 c*
1  c  tc  
From the expression for how the real exchange rate varies as a function of the
interest rate:
   0  (r  r f )
the equation for exports as a function of the exchange rate:
NX  X( ,Y f )  IM(Y)  X 0  xY f    Y
and the expression for the value of the multiplier:
Y 
A
A

1  c * 1  c  tc  
gives us a value for the extra change in equilibrium aggregate demand produced
by a change in interest rates through its effect on the exchange rate and on
international trade:
Y 
r


r
1 c * 1 c  tc  
Combine these two channels--the effect of interest rates acting through
Chapter 8
47
investment, and the effect of interest rates acting through international trade--to
figure out what the total slope of the IS curve is:
Y 
  r ( )  r
(   )


r
1 c*
1 c*
1  c  tc  
The position of the IS curve depends on all the determinants of aggregate demand.
Back in chapter 7 we arrived at a simplified form, using the multiplier, of the
equation for the total level of output Y:
Y
A
A

1  c * 1  c  tc  
Where the marginal propensity to spend c* is equal to c(1-t) - , is equal to the
marginal propensity to consume times the share of pre-tax income that becomes
after-tax income, minus the marginal propensity to import. And where the level
of autonomous spending A is equal to everything else that affects equilibrium
output:
A  c0  I  G  X
Gross exports X depends on the exchange rate:
X  X( ,Y f )  X 0  xY f   
Chapter 8
48
and the exchange rate in turn depends on the interest rate, so we can write gross
exports as a function of the interest rate:
X  X(  0 ,(r  r f ),Y f )  X0  xY f   ( 0   (r  r f ))
And then substitute the expressions for gross exports and for investment as a
function of the interest rate:
I  I(r)  I0  r
in to get a new expression for autonomous spending A:
A      r  xY f  r f   c0  I0  G  X 0   0 
Replacing autonomous spending A with its components, and rearranging
produces an algebraic expression for the IS curve:
Y
c0  I0  G X0  xY f
1  c  tc  

  r f   0 
1  c  tc  

   r
1 c  tc  
The final term shows the slope of the IS curve: the higher is the real interest rate
r, the lower is the equilibrium level of output. The first term shows the effect of
other domestic determinants of aggregate demand on the position of the IS
curve. Increases in either government purchases or the baseline levels of
consumption c0, or investment I0 will move the IS curve to the right--by an
Chapter 8
49
amount determined by the size of the multiplier 1/(1-c+tc+m). The middle term
shows the effects of what is going on in other countries on the location of the IS
curve--a change in either foreign incomes Yf, foreign interest rates rf, the baseline
demand for exports X0, or foreign exchange speculators' views of the
fundamental value of the exchange rate 0 will shift the IS curve as well.
Moving to and along the IS curve.
Moving to the IS curve.
What happens if the current levels of national product and interest rates are not
on the IS curve? If the economy is above the IS curve on the diagram, then
national product is higher than aggregate demand. That means inventories rise
rapidly and unexpectedly. So businesses cut back production. Employment,
national product, and national income fall.
If the economy is below the IS curve, aggregate demand is higher than national
income. Inventories fall. Firms try to expand production in order to meet
unexpectedly high demand. As they do, national product, employment, and
national income rise.
The process that pulls the economy back to the IS curve works slowly, over
months. Firms respond to increases in inventories by contracting (and to
Chapter 8
50
decreases in inventories by raising) production. The economy can stay off of the
IS curve for a substantial time, with inventories building up or falling. For a
whole year during 1990-91, inventory investment was low. For a year and a half
during 1994-95, some $50 billion more was produced than there was demand for.
[Figure: Moving to the IS curve]
Moving along the IS curve.
Changes in the level of the real interest rate r will move the economy either left
and up or right and down along the IS curve: a higher real interest rate r will
produce a lower level of aggregate demand. A lower level of the real interest rate
r will produce a higher level of aggregate demand.
Note that a number of changes in the economy can induce a move of the
economy along the IS curve. The interest rate that is graphed on the vertical axis
of the IS diagram is a long-term, risky, real interest rate--the interest rate that the
investment committees of corporations use to evaluate whether investment
projects have positive net present value or not.
Thus at least three kinds of changes can move the economy along the IS curve:

Changes in the risk premium. If lenders become more averse to risk, and
Chapter 8
51
demand a higher premium over the interest rates on safe assets like
government bonds, then the real interest rate will rise and the economy will
move up and to the left along the IS curve, lowering equilibrium output.

Changes in the term premium. If lenders expect that short-term real interest
rates will rise, there will be an unusually large positive gap between current
short-term rates and the long-term rates that govern the economy's position
on the IS curve.

Changes in the level of short-term safe real interest rates. Last--but perhaps most
important--changes in the level of short-term safe real interest rates will
(holding the term premium and the risk premium constant) move the
economy along the IS curve. This is the channel through which the central
bank can try to manage the business cycle. By cutting short-term safe real
interest rates the central bank can move the economy down and to the left
along the IS curve, and so boost the equilibrium level of output; by raising
short-term safe real interest rates the central bank can move the economy up
and to the right along the IS curve, and so reduce the equilibrium level of
output--all as long as shifts in the term premium or the risk premium do not
neutralize the central bank's actions
Chapter 8
52
Shifting the IS curve
The location of the IS curve.
Anything that affects planned expenditure will change the location of the IS
curve. Optimism that leads to an investment spending boom, or expansionary
fiscal policy that raises government purchases (or cuts taxes), will shift the IS
curve to the right.
Among the things that will shift the IS curve to the right--increasing national
product for any fixed level of interest rate--are: an increase in businesses' relative
optimism that makes them more willing to invest, an increase in consumers'
optimism about future incomes that leads them to increase consumption
spending, a decrease in taxes that gives consumers more disposable income, an
increase in government purchases, or an increase in net exports.
Conversely, anything that reduces aggregate demand on the income-expenditure
diagram shifts the lS curve to the left.
We can see the large variety of different forces that can affect the position of the
IS curve by looking back at the equation we derived for the IS curve.
Y
c0  I0  G X0  xY f
1  c  tc  

  r f   0 
1  c  tc  

   r
1 c  tc  
Chapter 8
53
Changes in consumer behavior.
Increases--or decreases--in consumers' perceptions of their permanent income
change the intercept c0 in the consumption function--and so change the location
of the IS curve, moving it right (if c0 increases) or left (if c0 decreases). The
amount by which a given change c0 affects the equilibrium level of production
depends on the size of the multiplier, and thus on the marginal propensity to
spend:
Y 
c0
1  c  tc  
[Figure: Shift in consumer sentiment and the IS curve]
Changes in the international environment
The location of the IS curve depends to a great extent on what is going on in the
outside world. Any of four kinds of changes in the outside world will affect the
location of the IS curve, and thus the equilibrium level of output for any given
level of the interest rate r. These four are, first, a change in foreign incomes Yf;
second, a change in foreign interest rates rf; third, a change in the baseline
demand for exports X0, and, fourth, a change in foreign exchange speculators'
views of the fundamental value of the exchange rate 0. All of these will shift the
Chapter 8
54
location of the IS curve.
X
Y 
0
 xY f  r f    0 
1 c  tc  
[Figure: Shift in the international environment and the IS curve]
Changes in expectations of future growth: Schumpeterian cycles.
Changes in government policy.
Anything that changes the level of government purchases will affect the position
of the IS curve, once more by an amount depending on the size of the multiplier:
Y 
G
1  c  tc  
[Figure: An increase in government purchases and the IS curve]
Another change in government policy that affects both the position and the slope
of the IS curve is a change in the tax rate t. You can show that a small change in
the tax rate of amount t will change the equilibrium level of output by:
Y 
c Yt 
1  c  tc  
Chapter 8
55
The (Yt) term--the level of output times the change in the tax rate--is just the
change in the level of tax revenue collected. It is a natural measure of the size of
the change, the analogue of the change in government purchases. The 1/(1c+tc+) term is also familiar: it is our so-often-seen multiplier. The minus sign
comes from the fact that an increase in tax rates decreases aggregate demand.
There is, however, one unusual thing about the change in taxes: the extra "c"
term--the marginal propensity to consume--in the numerator. This extra "c"
appears because a change in tax rates does not change aggregate demand
directly, but only indirectly because the change in tax rates shifts disposable
income and thus consumption spending.
[Figure: Change in tax rates and the IS curve]
Chapter summary
Main points
[To be written]
Chapter 8
56
Analytical exercises
[To be written]
Policy exercises
[To be written, and revised for each year within editions]