Unit 1: Inflation
Money Demand
We’ll start out with a model of ‘money demand’ and solve for the price level via
equili
ium in the money market.
P = price level
Y = real GDP
M = Nominal money supply
v = velocity = PY/M
Velocity refers to the number of times a unit of cu
ency moves through the
economy in a transaction.
The basic idea is that nominal GDP (P x Y) = M x v. Why must this be true?
Remember GDP is a flow variable that measures transactions that occur within a
geographic unit within a given period of time.
Unit 1: Inflation May, XXXXXXXXXX / 58
Money Demand
We’ll start by assuming monetary neutrality. That is, real variables (e.g. real
GDP) is unaffected by the money supply.
In fact, we’ll assume that Y is exogenous and constant. Why is this an OK
assumption when studying a hyper-inflation?
Money supply, M, is also exogenous and controlled directly/perfectly by the
monetary authority. Is this realistic?
We won’t assume anything about P...that’s what we’re trying to explain (changes
in P = inflation!) ... but we still need to model velocity, v. How we decide to
model v is the main way in which different models/explanations of inflation differ.
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Inflation in our model
We get inflation (growth in P) using the money market equili
ium condition:
πt+1 =
Pt+1
Pt
=
Mt+1
Mt
vt+1
vt
So inflation in the money demand model can either come from increases in the
money supply or increases in velocity.
The Quantity Theory of Money assumes that velocity is constant, so changes
in M feed directly into changes in P and therefore money growth is the only
explanation for growing prices.
Changes in velocity will be the way inflation expectations enter this model.
Unit 1: Inflation May, XXXXXXXXXX / 58
Where are the firms?
Wait...but who actually changes prices in the real economy? What’s actually
happening here?
This will become much more clear when we study the New-Keynesian model.
For now, you can think of a doubling in M or velocity as consumers coming to the
firm with double the purchasing power (say, within a single day!) and because the
firm can’t meet this new demand, it has no choice but to double it’s prices.
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Nominal interest rates and money demand
Velocity is likely not perfectly constant, even in the short run. How might
nominal interest rates affect velocity?
When nominal interest rates increase, the opportunity cost of holding liquid assets
(money) increases. Why?
When the opportunity cost of holding money balances goes up, this incentivizes
consumers to hold smaller money balances, meaning each unit of money they do
hold must work harder (circulate more). So velocity increases.
So we can write, vt(it)
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Recap: nominal interest rate and inflation
What is the relationship between the nominal interest rate and inflation
expectations?
If a lender charges a nominal rate of 4% and inflation ends up being 2%, what
eal rate are they actually receiving?
But lenders don’t know what inflation will be beforehand! So if a lender wants to
eceive a real rate r, and they expect inflation to be π, what nominal rate should
they demand?
it+1 = rt+1 + Et [πt+1]
Therefore, velocity is increasing in Et [πt+1] → vt+1(Et [πt+1])
Unit 1: Inflation May, XXXXXXXXXX / 58
Money supply vs. inflation expectations
Now we’re ready to consider the two competing explanations for hyper-inflations.
1 + πt+1 =
Mt+1
Mt
vt+1(Et+1[πt+2])
vt(Et [πt+1])
But how do we model expectations about inflation?
Two methods:
1 Adaptive expectations (backward looking, agents use previous periods’ data)
2 Rational expectations (agents use all available information)
Unit 1: Inflation May, XXXXXXXXXX / 58
Adaptive expectations
Adaptive expectations are relatively easy to understand:
Et [πt+1] = αt−1πt−1 + αt−2πt−2 + ...
Where
∑
αt = 1 (i.e. a weighted average of previous periods’ inflation rates)
Intuition: People in the economy look to the recent past to form thei
expectations about inflation.
Usually, we think that αt → 0 as t → ∞
Unit 1: Inflation May, XXXXXXXXXX / 58
Rational Expectations
Rational Expectations means that your expectations are model consistent (not
systematically wrong) in the sense that you take all available information.
This does not necessarily imply perfect foresight: i.e. Et [xt+1] ̸= xt+1 in every
period. Sometimes we think people are genuinely ‘shocked’ or surprised.
However, modeling randomness/shocks is difficult. How can we capture the
“idea” of a shock without wo
ying about modeling probability distributions ove
events?
A simple tool we can use is called an ‘MIT shock’ - we write down a
‘deterministic’ or ‘perfect foresight’ model where the (ex ante) probability of a
shock is essentially 0...but then a shock occurs!
Therefore it makes sense for agents to act as if the shock won’t happen, and only
change their behavior once it does.
Unit 1: Inflation May, XXXXXXXXXX / 58
Rational vs. Adaptive
What does RE get us?
Bottom line: It allows us to capture the idea that news about the future can form
expectations, not just past experience (adaptive expectations)
To see the difference between the two types of expectations, let’s see how they
operate in the model of inflation outlined above:
Let Mt+1/Mt = 1 + µt+1 ... the growth in the money supply
Suppose the government has been increasing the money supply at a steady
(low) rate, µ forever (or, if you like, as long as anyone can remember).
Before period t (so for periods, ... t-3, t-2, t-1), the probability of this
changing was extremely low (we’ll just say it’s essentially 0).
All of a sudden, a (super low probability) change in the money supply
occurs! They start printing money faster (µ̃ > µ)
Unit 1: Inflation May, XXXXXXXXXX / 58
Rational vs. Adaptive
Because money growth has been steady forever, it makes sense that (in both
models) inflation expectations - and therefore inflation - has just been determined
y money growth for t-1, t-2, t-3 etc... so πt−1 = πt−2 = πt−3... = µ
Why does this make sense?
1 + πt−1 =
Mt−1
Mt−2
vt−1(Et−1[πt ])
vt−2(Et−2[πt−1])
= 1 + µ
At time t, the government announces that Mt+1 = (1 + µ̃)Mt , and money will
grow at µ̃ from then on.
Question: What does each model of expectations predict about the path of
inflation?
Unit 1: Inflation May, XXXXXXXXXX / 58
Adaptive Expectations
1 + πt =
Mt
Mt−1
vt(Et [πt+1])
vt−1(Et−1[πt ])
= 1 + µ
To make it easy, let’s do a simple case where αt−1 = 1 and αt−j = 0 for all j.
So Et [πt+1] = πt−1
So what are Et−1[πt ] and Et [πt+1]? Just µ!
Therefore, what is 1 + πt? πt = µ because the increase in M has been announced
ut hasn’t occu
ed yet!
What about 1 + πt+1?
1 + πt+1 = (1 + µ̃)
vt+1(Et+1[πt+2])
vt(Et [πt+1])
= 1 + µ̃
Unit 1: Inflation May, XXXXXXXXXX / 58
Rational Expectations
Solving the RE version is a bit more complicated ...
Et−1[πt ] = µ just as in the AE case. Why?
But what about Et [πt+1]?
Remember that, starting in t+1, there will be no more shocks, so
Et+1[πt+1] = πt+2 (it’s a perfect foresight model after t+1)
To solve this, we’ll use a ‘guess and verify’ method.
Unit 1: Inflation May, XXXXXXXXXX / 58
Guess and Verify
We’ll guess that πt+j = µ̃ for j ≥ 1
Then Et+j [πt+j+1] = µ̃
1 + πt+j = (1 + µ̃)
vt+j(Et+j [πt+j+1])
vt+j−1(Et+j−1[πt+j ])
= 1 + µ̃
Success!
So now we have everything we need to solve for πt
1 + πt = (1 + µ)
vt(Et [πt+1])
vt−1(Et−1[πt ])
= (1 + µ)
vt+1(µ̃)
vt(µ)
1 + µ
Unit 1: Inflation May, XXXXXXXXXX / 58
What’s the intuition?
In the AE case, no one at time t anticipates the effect of the increase in the money
supply at t+1. Their expectations are firmly routed in the past, meaning there is
no increase in inflation until t+1 when the monetary increase actually occurs.
However, in the RE case, agents take into account news about future money
growth before the growth actually happens.
In this way, inflation can actually precede money growth and become a
‘self-fulfilling prophecy’.
Unit 1: Inflation May, XXXXXXXXXX / 58
Weimer Germany
Source: Uribe, Martin. Lecture: Hyper-inflations and Their Ends. (2017)
Unit 1: Inflation May, XXXXXXXXXX / 58
Seignoirage
Why do governments do this?! What does it mean to ‘monetize’ government
spending?
Suppose an economy is only capable of producing Ȳ .
If the government wants to re-apportion some of that output (Y’) - they could
just tax their population: T=Y’. Why might this be difficult?
Suppose instead that the government prints M units of money and uses this new
money to buy Y ′ instead.
How much will they have to print? Because the new money will increase the price
level, they’ll have to print more than Y’ in order to siphon off Y’ in real output.
Unit 1: Inflation May, XXXXXXXXXX / 58
Seignoirage Revenue
How to solve for M’, the amount of new money they’ll have to print to get Y’?
Note that the following must be true:
M0 +M
′
Pnew
= Ȳ
1
v
M ′
Pnew
= Y ′
1
v
We have 2 equations and 2 unknowns (M’ and Pnew)
So M0Pnew = (Ȳ − Y
′) 1v →
vM0
(Ȳ−Y ′) = Pnew
Finally, M ′ = Y ′ 1v ∗ Pnew
Unit 1: Inflation May, XXXXXXXXXX / 58