EC201 Intermediate Macroeconomics
EC201 Intermediate Macroeconomics
Problem set 4 Solution
1) (Money demand and Fisher effect) Suppose that the money demand in an economy
is given by the following linear function:
M
= 500 + 0.2Y − 1000i
P
a) Suppose that P=100, Y=1000 and i=0.1. Determine the demand for real
balances and the velocity of money in this economy;
b) Suppose that now P=200 while everything else remains unchanged. Determine
the new demand for real balances and the new velocity in this economy;
c) Given the previous money demand function, determine how the velocity of
money is affected by an increase in real income in this economy. Determine
how the velocity would be affected by an increase in inflation;
Solution
a) Given the data in the problem we have:
M
= 500 + 0.2 × (1000) − 1000 × (0.1)
P
M
⇒
= 600
P
and total money supply is: M = 600 × 100 = 60000
The velocity of money according the quantity theory is given by:
Y
Y
1000
V =P
=
=
= 1.66
M M / P 600
b)
Now the price doubles. However, we assume that everything else remains unchanged,
meaning that the real Income and the nominal interest rate do not change (this is a big
assumption since we know that the nominal interest rate depends on inflation, but it is
like people do expect that inflation will not change so r + π e does not change).
In this case the demand for real balance must be the same since nothing has changed
apart the prices:
M
= 500 + 0.2 × (1000) − 1000 × (0.1)
P
M
⇒
= 600
P
In this case the total money supply simply has double as well:
M = 600 × 200 = 120000
c)
The velocity of money is:
Y
Y
Y
V =P
=
=
M M / P (500 + 0.2Y − 1000i )
First: the velocity does not depend on the price level as long as the nominal interest
rate does not change (meaning: the expected inflation does not change).
However, we can see what the effect of a change in expected inflation on the velocity
of money is (this is not the same as looking for the effect of a change in the price
level!).
∂V
1000Y
=
>0
e
∂π
(500 + 0.2Y − 1000(r + π e )) 2
As far as a change in income is concerned:
∂V
500 + 0.2Y − 1000i − 0.2Y
500 − 1000i
=
=
e
2
∂Y (500 + 0.2Y − 1000(r + π ))
(500 + 0.2Y − 1000(r + π e )) 2
If the numerator is positive, then that derivative is positive as well.
Therefore:
∂V
if 500 − 1000i > 0 ⇒ i < 0.5
>0
∂Y
This means that the velocity of money will increase after an increase in real income if
the interest rate is lower than 50%.
2) (Baumol-Tobin money demand) The Baumol-Tobin money demand is given by:
M
γY
=
P
2i
where Y is real income, γ is the cost of inconvenience of going to the bank to
withdraw money and i is the nominal interest rate. This was derived in Lecture 4.
Moreover the optimal number of times ( f ) that an individual should go to the bank in
a given period is given by:
Yi
f =
2γ
a) Suppose that the interest rate is 10% (=0.1), the cost of each trip to the bank is
£25 and the annual expenditure Y is £72000. How many times in a year
should we go to the bank according to the Baumol-Tobin model? How many
times in average per month? How much should we withdraw each time? What
is the average money holding?
b) According to the Baumol-Tobin money demand, what is the elasticity of
money demand with respect to income? What about the elasticity with respect
to the interest rate? (Hint: rewrite the money demand using logs).
c) Empirical evidence shows that the elasticity of money demand to income is
close to 1. How would you modify the Baumol-Tobin money demand to be
consistent with this finding?
Solution
a) Using the data of our problem into the equation for f , we have:
72000 × 0.1
7200
=
= 144 = 12
2 × 25
50
In a given year we should go 12 times to the bank. This means that in average we go
once to the bank each month.
Since we have an expenditure of £72000, each time we go to the bank we should
withdraw £72000/12=£6000
The average money holding each month is clearly £6000/2=£3000 (half of the time
we hold £6000 and half of the time we hold £0).
f =
Of course we can use the money demand equation to find the average money holding,
since that expression is exactly the average money holding an individual demand:
M
25 × 72000
=
= 9000000 = 3000
P
2 × 0.1
b)
Using the logs (for example the natural log or the log in base 10):
 γY 
M 

ln  = ln

2
i
P


We need the following properties of logs:
 A
ln  = ln( A) − ln( B )
B
ln( AB ) = ln( A) + ln( B )
ln( A B ) = B ln( A)
Using those properties we have (we keep the left hand side as the log of a ratio):
1
1
1
M  1
ln  = ln(γ ) + ln(Y ) − ln(2) − ln(i )
2
2
2
P 2
Given two variables related by a function like Y = f ( X ) the elasticity of Y with
respect to X is defined as:
dY X
εY ,X =
dX Y
This is equivalent to
d ln(Y )
ε Y,X =
d ln( X )
Therefore the elasticity of money demand with respect to income can be found as:
d ln(M / P) 1
ε M / P ,Y =
=
d ln(Y )
2
This explains why using logs may be very useful in economics. Elasticities can be
found quite simply when a model is expressed in logs.
According to the Baumol-Tobin model if income (or expenditure) increases by £1,
average money holding will increase by £0.5.
The elasticity of money demand with respect to the interest rate is defined similarly
as:
d ln( M / P)
1
ε M / P ,i =
=−
d ln(i)
2
c)
To make the elasticity of money demand with respect to income we need a way to
have Y 2 under the square root formula.
A possible way, with some economic meaning, of doing that is to assume that the cost
of going to the bank depends linearly with income. Suppose for example that γ = bY
where b > 0 is a constant. Assume that b < 1 . Then γ is just the opportunity cost in
terms of real income. Suppose that real income is £15 per hour. Going to the bank
takes 20 minutes. So the cost opportunity of going to the bank in terms of real income
is £15/3=£5 and b=1/3. If your real income increases, the cost opportunity of going to
the bank increases because your time becomes more valuable.
With this assumption the Baumol-Tobin money demand is then:
M
bY 2
=
P
2i
Using logs:
1
1
M  1
ln  = ln(b) + ln(Y ) − ln(2) − ln(i)
2
2
P 2
Therefore the elasticity of money demand with respect to income is now equal to 1
and it is more consistent with the empirical evidence.
3) (Cagan’s model of Seigniorage). Consider an economy in which the money
demand function is given by:
M
= Ye −α ( r +π )
P
where Y is real income, r is the real interest rate, π is the rate of inflation and α >0 is
a constant. Define the revenues from seigniorage as:
M
S =σ
P
dM
where σ is the growth rate of money supply (=
).
M
Now assume that σ = π .
a) Write down the revenues from seigniorage as a function of real income, the
real interest rate and the rate of inflation. (Hint: don’t use logs).
b) Given the function you found in a), find the rate of inflation that maximises
the revenues from seigniorage. Explain your results. (Hint: don’t use logs.
Uuse the fact that: given a function y = e f ( x ) , its derivative is
dy df ( x) f ( x )
=
e )
dx
dx
Solution
a)
In the 50s Cagan tried to answer the following question: if seignorage is an inflation
tax and a government can control the inflation rate by controlling (through the central
bank) the growth rate of money supply, what is the inflation rate that a government
should choose in order to maximise revenues from signorage?
Using the definition of seigniorage and the money demand function the revenues from
seigniorage are:
M
S =σ
= πYe −α ( r +π )
P
where we used the fact that σ = π .
b)
We need to maximise S with respect to π :
∂S
= Ye −α ( r +π ) − απYe −α ( r +π ) = (1 − απ )Ye −α ( r +π )
∂π
Set that derivative equal to zero and find the value of π that solves that:
∂S
= (1 − απ )Ye −α ( r +π ) = 0
∂π
It is clear that the equation above is equal to zero when π =
1
α
.
Therefore the seigniorage revenues are maximised when the inflation rate is π =
1
α
.
α measures the sensitivity of the demand for real balances with respect to the
nominal interest rate. Higher is that sensitivity and lower must be the inflation rate
that maximises the inflation tax revenues.
Intuition: suppose that α is large. If the government increases money supply to
increase the inflation tax it will also increase the nominal interest rate through the
Fisher effect (i = r + π ) since inflation will rise. If α is large, money demand will
decrease quite a lot if the nominal interest rate increases. Since inflation tax is a tax on
the money that people hold, if demand for money decreases, also the taxable base
decreases (people hold less money) and therefore the revenues from the inflation tax
will not increase very much.
Notice that when inflation is zero then S = 0. Therefore, Seigniorage revenues
1
increases in the inflation rate until π = . After that point (since it is a maximum
α
point) S must decrease as inflation continues to increase.
Therefore, as for any other tax, there is a Laffer curve that describes the tax revenues
as a function of the tax rate. The Laffer curve gives us an idea of thade-off between
increasing the tax rates and increasing the tax revenues. The classical example is the
taxation on income. The idea is that when the tax rate is zero the tax revenues are zero
as well, but also if the tax rate is 100% the tax revenues will be zero since nobody will
have an incentive to work since all their income will go to tax revenues. Therefore,
there should be a point between 0 and 100% where the tax revenues are maximised.
The same applies to the inflation tax and the Laffer curve in this case is the following
Graphically:
S
S*
S1
π1
π=1/α
π2
π
In the picture we have that S is first increasing in the inflation rate but then is
decreasing since people is getting rid of their money faster than the inflation and
therefore the taxable base is decreasing faster than the tax rate (the inflation rate) and
therefore revenues are decreasing.
Notice that, apart where the inflation rate maximises the seigniorage revenues, there
are always two different inflation rates that give you the same amount of tax revenues.
For example, we can obtain S1 with an inflation rate π 1 or with π 2 .
This means that if we are on the right-hand side of π = 1 / α we are always in an
inefficient situation. We can decrease the inflation rate and at the same time
increasing inflation tax revenues.