Notesale: Turn your study into money

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EC201 Intermediate Macroeconomics
EC201 Intermediate Macroeconomics

Lecture 4: Money and Inflation contd
...
4 and Ch
...
We have seen money supply now we turn to money demand
...
etc,)
...

By liquidity of an asset we mean the ease in which that asset is converted into other
goods
...
A bond is less liquid because you
need to wait a certain period of time to have back the money with the interest rate on
it before you spend it
...
Those are provided by Keynes and will be
discussed again in detail in Lecture 6
...
The reason
we may keep cash is that there is no synchronisation between when we get
paid and when we need to buy goods;
b) Precautionary motive: we may keep money in cash form not just to buy goods
but also as a cushion against unexpected events;
c) Speculative motive: people may hold money in cash form to speculate on the
future prices of bonds and other assets;
If people desire to hold money (for the motives above) then there is a demand for it
...
Therefore they should be related to
the level of transactions
...

Motive c) is related instead with the cost opportunity of holding money that is the
nominal interest rate i
...

Hence, ↑i ⇒ ↓ in money demand
...

The other assets typically generate some type of income (e
...
interest income in the
case of bonds), but are much less liquid than money
...

With this for background, an individual’s “money demand” refers to the fraction of
his wealth he would like to hold in the form of money (as opposed to less-liquid
income-generating assets like bonds)
...

Therefore, we can assume an aggregate money demand function like the following
one:
d

M 
  = L (Y , i )
 P

1)

where L is a function that relates the demand for real balances (M/P) with the real
income (Y) and the nominal interest rate (i)
...

M/P = real money balances, the purchasing power of the money supply
...
Therefore the demand of
money is a demand for real money balances
...
This is
because the nominal interest rate is the opportunity cost of holding money as we
explained earlier
...
In this example the function L is a linear
function
...
Hence, the nominal interest rate relevant
for money demand is r + π e (see previous lecture note about the Fisher effect)
...
The supply of real money balances is therefore
given by:
s

M  M
  =
P
 P

3)

Notice that since the supply of money is exogenous it does not depend on any
variable
...
You DO need
to know Y in order to determine L, and you need to know L and M in order to
determine P
Therefore equation 4) tells us how the general level of prices is determined in the
long-run
...
The quantity theory we analysed in the previous lecture gave us a similar
idea about the way prices are determined in the long run
...

Consider equation 4) and suppose that the central bank increases the money supply M
while the demand remains unchanged
...
Therefore, also equation 4) implies
that the price level can be proportional to the money supply in the long-run (given the
demand for money)
...
Since we have a demand and a supply of
money we have also a price for money
...

An example of an estimated money demand
d

M 
Consider the money demand   = aY − bi
...
As an example, I did collect annual data for the US on M2 (the stock of
money), the CPI index for the general price level, the Real GDP and the 3-month
treasury bill rate for the nominal interest rate
...
etc
...
362 LnRGDP − 0
...
006)

4

(0
...
The sample is from
1959 to 2009
...
This is in line with the assumptions we made on the money demand
function L
...
After
1980s money demand estimates were not giving good results because money demand
became more unstable
...
etc
...
Here we derived a money
demand similar to the previous one but now using a microfounded model in which
money demand results as the optimal choice of rational agents
...
The
demand function we derive is called the Baumol-Tobin money demand
...
Bank deposits pay a positive interest rate i while cash does not
...

Problem: if you hold only bank deposits you need to go the bank (or to the ATM) to
withdraw cash quite often and that is inconvenient
...
On the other hand if you hold only cash you
lose the interest payments you can get from a bank deposit
...
etc
...

Denote with M w the amount withdraw each time the individual goes to the bank
...


5

An example of what is going on is in the next figure:

In average, each period, you hold an amount of money M =

Mw
(half of the time in
2

each period you hold M w and half of the time you hold 0)
...
Denote with Y the quantity of goods the individual
can buy and with P the price level of those goods
...

Using M =

Mw
→ 2M = M w , equation 5) can be written as:
2
2 Mf = PY

6)

This represents the budget constraint of our representative individual
...
It is inconvenient to go often to the bank
...
Your hourly wage is £12, then the
cost of going to the bank is £4)
...

Moreover holding cash is costly because you lose the interest rate payments on the
amount of cash you hold
...
Remember that what matters for the
P

individual in the purchasing power of the money holding
...

So in this model f and m are the endogenous variables while all the others are
exogenous variables
...
Therefore equation 8) can be written as:

2mf = Y

9)

The constrained minimisation problem faced by the individual can be solved in
various ways, here we just substitute the constraint in 9) into the cost function in 7) to
get rid of the constraint
...

2m

Substitute that into 7) to get:

TC =

γY
2m

+ im

10)

Now we can ask: what is the value of m that minimise the total cost in 10)?
Take the first order derivative of 10) and set it equal to zero and then solve for m (we
assume that the second order conditions for a minimum are satisfied so we will not
check them):

dTC
2γY
=−
+i = 0
dm
4m 2
⇒ −γY + 2m 2 i = 0
⇒ m2 =

γY
2i

Therefore the optimal real money balances holding is given by:

m≡

M
γY
=
P
2i

11)

Notice that we have taken only the positive root because the negative root (a negative
amount of money holding) has no economic sense
...
Moreover it depends negatively on
the interest rate ( i )
...
It represents a
 P
microfoundation of it
...
If you do it, after a bit of algebra you
should find: f =

Yi

...
Equation 11) is the average money holding in
each period
...
Financial
innovation is related to that
...


Money Market and Expected Inflation
Over the long run, people don’t consistently over- or under-forecast inflation, then

π e = π on average (this does not mean that nobody makes mistakes in the long-run, it
means that in average the expectations of people are correct)
...

For example: suppose the Bank of England announces it will increase M next year
...

This affects P now, even though M hasn’t changed yet
...

How P responds to ∆πe
For given values of r, Y, and M :

↑ π e ⇒↑ i (the Fisher Effect)

M 
↑ i ⇒↓  
 P
↑ P ⇒↓

d

M M 
M
= 
to re-establish the equilibrium:
P
P  P

8

d

The idea is the following: at time t everything is kept constant; the only thing that is
happening is the central bank announcing that at t +1 the stock of money will be
higher than the one at time t
...
The
increased nominal interest rate decreases the money demand at t
...
If
expectations are considered in the analysis, policy announcements about the future
can have effects today
...
From the previous reasoning we know that the sign of that
∂π e

derivative should be positive
...
Rewrite equation 4) as:
P=

M
L(Y , r + π e )

Then:

∂L ∂i
−
∂P
∂i ∂π e
=M
2
e
∂π
L(Y , r + π e )

[

To obtain

]

12)

∂L
we need to use the chain rule (see the Appendix)
...

∂π e ∂i ∂π e

This explains the expression in the numerator
...
Furthermore
= 1 , therefore, the
∂i
∂π e

numerator of expression 5) is positive
...
Therefore

∂P
> 0 , an increase in the expected inflation will
∂π e

lead to an increase in the general level of prices everything else constant
...
4 of the Mankiw’s
book)
...
It is an exponential
function, as indicated by the presence of the number e, in the nominal interest rate
...

Equation 13) is not very useful as it stands; however, if we take natural logs of 13) we
obtain the following more tractable demand function
ln(M t ) − ln(Pt ) = ln(Yt ) − αit

14)

(see the appendix for the properties of logs)
...
This explains the differences in the notation)
...
Therefore we can write

π t +1 = ln(Pt +1 ) − ln(Pt )
...

The expectation applies only to the variables that are not known at time t
...

Using the lower case to denote variables in log we can write equation 14) as:
mt − pt = yt − αit

15)

This is just the log version of equation 13)
...
Now assume for simplicity that r and y are normalised to zero and
delete them from that equation
...

Solve equation 17) for pt
pt =

α
1+α

Et pt +1 +

mt
1+α

18)

Equation 18) must hold for any time period t, therefore at time t+1 it must be true
that:

pt +1 =

α
m
Et +1 pt + 2 + t +1
1+ α
1+α

19)

at time t+2:
pt + 2 =

α
m
Et + 2 pt + 3 + t + 2
1+α
1+ α

20)

and so on
...
It can be shown that Et Et +1 pt + 2 = Et pt + 2 by the law of

iterated expectations
...
We can then substitute the expression for pt+2 from 20) into 21)
...

mt +
1+α
1+α 





11

We now assume that 0 <

α
< 1
...

1+ α 
n

This means that for n large enough the first term in the right-hand side vanishes
...

mt +
1+α 
1+ α
1+ α 




22)

Equation 22) tells you something important: the current level of prices pt depends on
not only the current but also the entire expected future path of the money stock
...
Suppose that
the central bank that controls the money supply, can credibly commit to announce
today that: from now on the money supply will be kept stable at some particular level
...

However, assume that the central bank is not credible and people believe that at some
point in time in the future the money supply will be increased more than it was
announced, then people will adapt their expectations today and the price level
TODAY will increase even if no increases have occurred in the money supply yet
...


costs when inflation is expected: for example we expect inflation to rise 3%

every year in the future
...


costs when inflation is different than people had expected

1) The costs of expected inflation
a) Shoeleather cost
definition:

the costs and inconveniences of reducing money balances to avoid

inflation
...
6 and suppose that t denotes a particular year
...
Then 


1+ α 

12

10

= 0
...


⇒ ↓ real money balances
Remember: In long run, inflation does not affect real income or real spending
...

b) Menu costs
definition: the costs of changing prices
...

c) Unfair tax treatment
Some taxes are not adjusted to account for inflation, such as the capital gains tax
...

Suppose π = 10% during the year
...

But the government requires you to pay taxes on your $1000 nominal gain
...
Suppose you have a wage of £30000, and
above £10000 you must pay an income tax of 20%
...
Therefore, the income tax you have to pay is:
(30000-10000)0
...
In practice you pay (4000/30000)100 = 13
...
Now suppose that there is a 10% increase in the inflation and you
wage increases by 10%
...
However, assume that the threshold is not adjusted and it is still
£10000
...
2 = £4600
Now you pay 4600/33000 = 13
...

The proportion of income you pay as taxes is increased and this is called fiscal drag
...
This is a problem especially when inflation is particularly high and with

13

indirect taxes (such as value added taxes)
...
If inflation is high,
there will be a deterioration of the real value of taxes collected by the government
...
g
...
If price level is particularly high at t+1
compared to t, the government is loosing some real income from those taxes
...

This complicates long-range financial planning
...

- Workers trying to decide how much to save for retirement
...


2) The costs of unexpected inflation
a) Arbitrary redistribution of purchasing power
Many long-term contracts not indexed, but based on πe
...

Example: borrowers and lenders
...
You need to repay the amount borrowed plus a nominal interest rate i
...

The nominal interest rate depends on the inflation expected at time t+1
...
The
borrower repays the loan with less valuable dollars for example
...

For example, at time t you borrow $1000 and you agree to pay back $1000 plus a
nominal interest rate of 10% at time t+1
...
This means that the lender expect an inflation rate of 5% at t+1
...

At t+1 you pay back $1000 + $100 = $1100
...
15= $1150
...
On the other hand the borrower is better-off
...
Arbitrary redistributions of wealth become more likely
...

Seigniorage and Hyperinflation
Government collects taxes
...

To spend more without raising taxes or selling bonds, the government can print
money (we normally say that it is the central bank that prints money, but in many
countries the central bank is still under the direct control of the government)
...

The inflation tax: Printing money to raise revenue causes inflation
...
Why is inflation similar to a tax?
Suppose I have $500 in cash in my pocket when the government suddenly announces
it has printed up enough extra dollar bills to double the economy’s cash supply
...
The $500 in
my pocket will buy only $250 would have bought before
...

Where did the $250 real dollars in my pocket go? The government has them: it now
has 500 newly-printed dollars, even if each of them is worth half a pre-inflation dollar
in real terms
...

To collect an explicit tax a government needs need an entire wealth-tracking, moneycollecting, and compliance-monitoring bureaucracy
...
Moreover, the inflation tax is a heavy tax on one small slice of
money-holding, mainly cash
...
S
...
In
Italy and Greece, seigniorage has often been more than 10% of total revenue
...

What is hyperinflation?
We define hyperinflation a situation where the inflation rate exceeds 50% per month
...
Money ceases to function as a store of value, and may not serve its
other functions (unit of account, medium of exchange)
...
Hyperinflation is caused by
excessive money supply growth
...
This reduces the demand for money by raising the nominal interest rate (through
the Fisher effect)
...
Some examples of hyperinflation:
money

growth

(%)

inflation (%)

Israel, 1983-85

295

275

Poland, 1989-90

344

400

Brazil, 1987-94

1350

1323

Argentina, 1988-90

1264

1912

Peru, 1988-90

2974

3849

Nicaragua, 1987-91

4991

5261

Bolivia, 1984-85

4208

6515

In the table above we have a close relationship between money growth and growth in
the general price level (inflation)
...


16

Highest Monthly Inflation Rates in History

Country

Time
Month with
Equivalent
required
highest
Highest monthly
daily
for
inflation
inflation rate
inflation
prices to
rate
rate
double
195%

15
...
0%
2008 (latest
measurable)

24
...
30 x 1016%

Yugoslavia

January
1994

313,000,000%

64
...
4 days

Germany

October
1923

29,500%

20
...
7 days

Greece

November
1944

11,300%

17
...
5 days

China

May 1949

4,210%

13
...
6 days

Source: Prof
...
Hanke, February 5, 2009
...

In theory, the solution to hyperinflation is simple: stop printing money
...


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Mathematical Appendix
1) The Chain Rule of basic calculus:
Consider a function f (x) where x = g (z )
...
That
dz

derivative is given by:

df df dx
=
dz dx dz
For example: f ( x) = x 2 and x = z − 1
...
Therefore:
= 2 x and
dx
dz
df
= 2x
dz

Using the definition of x into the above equation we obtain:
df
= 2( z − 1)
dz
In this simple case we could have substituted the definition of x directly into f(x)
...
Calculating directly the derivative

df
gives
dz

you exactly the same result as before
...

Consider for example a general function of two variables: f ( x, y )
Assume that y = 3z + 2w
Suppose you want to calculate

∂f
(here we use the notation of partial differentiation
∂z

∂ since we deal with a function with more than one variable)
According to the chain rule, and knowing that

∂y
= 3 we have:
∂z

∂f ∂f ∂y
∂f
=
=3
∂z ∂y ∂z
∂y

2) Properties of the logs
The log with base b of a number y is defined as

18

logb y = x
It gives a number x that is the exponent of the base b that gives you the number y:

y = bx
There are many bases we can use, however, base 10 and base e are the most common
...
718…
Logarithms with base e are called natural logarithms and are denoted with ln
...

c) ln( AB) = ln( A) + ln(B)

( )

d) ln Aα = α ln( A)
where α can be any number or a function
...
1) consider a function y = ln x
dy d ln( x ) 1
≡
=
dx
dx
x
f
...

dx

f
...
Take natural logs on both sides, this gives
you: ln( y) = ln( f ( x))
...
Suppose you want to calculate

d ln( y )

...


19

--------------------------------------------------------------------Consider the demand function 6) in the lecture note:
Mt
= Yt e −αit
Pt

Taking natural logs on both sides:
M
ln t
 P
 t

(


 = ln Yt e −αit



)

Using the properties of the logs we have for the left-hand side:

M 
ln t  = ln(M t ) − ln( Pt )
 P 
 t 
In the right-hand side we have a product of two terms, Yt and

(

e −αi
...


Why do we take logs of a function?
Because it is sometimes easier to work with the logarithmic version of a function than
with the original function itself
...
Therefore, we do not affect the
substance of the function with this operation
...

By taking the log of that function we still have that ln(f(a))>ln(f(b))
...

Another advantage of taking logs of a function is that the parameters of that function
can now be interpreted as an elasticity (from property g)), and in economics we tend
to like elasticity
...


Logarithms and the growth rate of a variable:
From property f
...
This
x

means that instantaneous growth rate of x is equal to the absolute change in the
natural log of that variable d ln(x)
...
That is:

∆ ln( x) ≅
where the symbol

∆x
x

A1)

≅ means approximately equal
...
2
x2000
10
and so on…
As you can see the growth rates calculated in the two different ways are
approximately similar
...

Consider the inflation rate in period t:

πt =

Pt − Pt −1
Pt −1

This can be approximated using the logs:

π t ≅ ln(Pt ) − ln(Pt −1 )
In many cases instead of using the symbol

≅ we will assume that the approximation

is good enough to use the symbol =
...


Another useful approximation using logs
Suppose that x is a positive number but quite small, then:
ln(1 + x) ≅ x

For example, suppose x=0
...
1) = 0
...
1
...

The Fisher equation comes from the following equation:
(1 + rt ) =

(1 + it ) Pt
Pt +1

Taking the logs on both sides:

ln(1 + rt ) = ln(1 + it ) + ln Pt − ln Pt +1

A2)

Using the approximation we have: ln(1 + rt ) ≅ rt and ln(1 + it ) ≅ it
...

So equation A2) can be approximately written as:

22

rt = it − π t +1
Or using the notation in the lecture note:

r = i −π e

23



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