Modelling expectations
• In this lecture we will complete the transition and
develop a model in growth- inflation, output
growth, money stock growth and unemployment.
• In our main AD-AS model our solution depends
on expectation about the future. These
expectations come through the labour wagesetting equation.
• As a result, the solution to our model in the
present requires us to make some assumption
about how people form expectations of the
future. The present depends on peoples’ beliefs
about the future.
Modelling expectations
• We have already seen a few ways to model how
people form expectations in our Phillips curve
equation
πt = θπt-1 + (μ + z) - α ut
– θ = 0 - expectations never change
– θ = 1 - tomorrow will be just like today
• “Adaptive expectations”: assumed that people
adjusted their expectations partly based on what
they experienced today- 0 < θ < 1
Modelling expectations
• Another form of expectations is “rational
expectations”. In this case, we assume
that people “know” our model of the
economy and form their expectations
based on the “true” model of the economy
and the information they have available.
– Intuition: Imagine that economists have a
decent model of the economy. We would
expect that people would check their
expectations against this model.
Example of rational expectations
• Imagine AD is higher
than expected, SR
equilibrium is A.
• Wage demands push
wages and prices up
to A’, slowly up to A’’.
• But imagine workers
“know” this model,
instead they demand
A’’, and we move
straight to A’’.
How else do expectations matter?
• Investment: When you make an
investment, you are always making some
guesses about the future, as investments
are costs incurred today in the hope of
some gains in the future.
• In this lecture we will be looking at
introducing expectations into our
investment and consumption decisions.
Nominal vs. real interest
• We already know about deriving “real GDP”- that
is a measure of GDP that does not depend on
changing prices.
• But investment decisions face the same problem
with respect to prices. I buy the investment
good today at today’s prices, but the investment
good pays off in the future at future prices.
• We need to define a real interest rate that nets
out the effect of price changes.
Real interest rate
• Here what we are imagining is a rate of return
on an investment that is not affected by price
changes.
• “Rate of return”: the rate of return is the value of
the investment (V) next year divided by the cost
of the investment this year
(1 + it) = Vet+1 / Vt
• Imagine we have an investment good that costs
Pt and returns (1 + rt) investment goods next
year.
Real interest rate
• The investment good costs Pt this year
and will return Pet+1(1+rt) next year, so our
return is:
(1 + it) = (1+ rt)Pet+1 / Pt
• Rewriting this becomes:
(1 + rt) = (1+ it) Pt / Pet+1
• And defining expected inflation by:
πet = Pet+1 - Pt / Pt
Real interest rate
Pt / Pet+1 = 1 / (1 + πet )
• And so we derive:
(1 + rt) = (1+ it)/(1+ πet )
• As log x is approximately equal to 1+x for
small x, then:
rt ≈ it - πet
• Intuition: The nominal interest rate is
made up of a real return plus an inflation
term.
Brief review of discounting
• Any investment decision involves comparing
some cost now to some benefit in the future.
• But $1 today is not the same as $1 in the future.
• The question then is: what is $1 in the future
worth?
– One answer is to say “how much would I have to put
in the bank now to have $1 in the future?”
• $1 in the bank today becomes $(1+it) next year
through interest payments.
Discounting for time
• $x in the bank becomes $(1+it)x next year.
• If I want to have $1 next year, I need to put $x in
the bank:
$(1+it)x = $1
x = 1/(1+it)
• If I want to have $1 in two years’ time, I need to
put $x in the bank:
$(1+it)(1+iet+1)x = $1
x = 1/(1+it)(1+iet+1)
• Where iet+1 is the expected rate of interest 1 year
from now.
Valuation principles
• Imagine we’re thinking of placing a monetary
value on an asset. Imagine that people only
value the flow of cash they expect to gain from
the asset, so the asset has no “consumption
value”, unlike a painting or a sports car.
• We can then value the asset by valuing the cash
flow of the asset. People are indifferent between
holding the asset and holding the right to an
identical cash flow.
• But the cash flow is in the future, so we need to
discount the cash flow and get the “present
value”.
Valuation of an asset
• Imagine an asset is expected to return in
cash $zet+1 next year, $zet+2 the year after
next, $zet+3 the year after that …
• The value of the asset is the sum of the
discounted values of these cash flows:
Vt = $zet+1/(1+it) + $zet+2/(1+it)(1+iet+1) + …
• This is the basis of financial valuation and
the basic tool of finance.
A bit of discounting maths
• An annuity is an asset that pays a
constant flow of cash over time. What is
the value of an annuity? Imagine the
annuity pays $z every year forever and
imagine that we expect interest rates to be
i forever.
V = $z/(1+i) + $z/(1+i)2 + …
V = $z/(1+i) [1 + 1/(1+i) + 1/(1+i)2 + …]
A bit of discounting maths
• But we know that
1 + s + s2 + s3 + … = 1/(1-s)
• [Hint: Define S = 1 + s + s2 + s3 + …, then
you will see that S – 1 = sS. Rewrite to
get result.]
• If s = 1/(1+i), then 1/(1-s) = (1+i)/i.
V = $z/(1+i) [(1+i)/i] = $z/i
• The value of an annuity that pays $z
forever is $z/i.
A bit of discounting maths
• What if the annuity only lasts n years?
• The easiest way to think about this case is
that the value of an n year annuity is equal
to the infinite annuity minus the resale
value of the annuity after n years.
V = $z/i – 1/(1+i)n+1($z/i)
V = ($z/i)[1 – 1/(1+i) n+1]
Value of a share
• Example: The price of a share should be equal
to its cash flow value. Imagine we buy a share
today at price Pt and sell it next year at price
Pet+1. In the meantime we get the expected
dividend next year det+1.
Pt = det+1/(1+it) + Pet+1/(1+it)
• But the expected value of P next year must be
the expected value of the dividend plus the sale
price the year after next.
Value of a share
• So we get
Pt = det+1/(1+it) + [det+2/(1+iet+1) + Pet+2/(1+iet+1)]
/(1+it)
• We can keep doing this and finally get:
Pt = det+1/(1+it) + det+2/(1+iet+1)(1+it) + …
• The value of a share must be equal to the
present value of the expected dividends from
the share.
• Share prices and dividends and expected
dividends should move in the same direction.
Price-earnings ratio
• How can we use these equations to help us
understand the stock market?
• A share is exactly the same as an annuity. The
value of the share comes from the discounted
(possibly)-infinite stream of dividends. So if we
have a share with value V and if dividends are
constant at z, then we should have:
V = z/i or 1/i = V/z
• In the market V is called the “price” and z is the
“earnings”, then the price-earnings ratio should
be 1/i.
Value of a bond
• Imagine a piece of paper that says “I will pay the
holder of this piece of paper $100 in 1 year’s
time. Signed, XXX.” XXX could be a company
or a government.
• These pieces of paper are called “bonds”. This
is a “discount bond” because the payment is a
flat $100 and so the price of the bond will be a
discounted value of $100.
• What is the value of a promise of $100?
Value of a bond
• If we don’t have to worry about a broken
promise, the value of $100 in one year is:
P1t = $100/(1+it)
• The value of $100 in two year’s time is:
P2t = $100/(1+it)(1+iet+1)
• We see that bond prices and interest rates and
future interest rates move in opposite directions.
So if the financial community predicts interest
rate rises, we should see bond prices fall.
Value of a bond
• In the financial markets, however, we don’t see
interest rates, but instead we see bond prices.
Can we go from bond prices to interest rates?
Yes.
• We see P1t and we know the face value of
discount bonds, so we can use:
1+it = $100/P1t
• A fall in bond prices means that the financial
community expects interest rates to rise.