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 wage-setting 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 Pe

t+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 = Pe

t+1 - Pt / Pt

Real interest rate

Pt / Pet+1 = 1 / (1 + πe

t )• 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 = $1x = 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 $ze

t+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) + $ze

t+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 that1 + 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 Pe

t+1. In the meantime we get the expected dividend next year de

t+1.

Pt = det+1/(1+it) + Pe

t+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) + [de

t+2/(1+iet+1) + Pet+2/(1+iet+1)] /(1+it)

• We can keep doing this and finally get:

Pt = det+1/(1+it) + de

t+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.