EC201 Intermediate Macroeconomics EC201 Intermediate

EC201 Intermediate Macroeconomics
EC201 Intermediate Macroeconomics
Problem Set 3 Solution
1) (Quantity theory of money and money demand)
a) Derive a demand function for real money balances from the above quantity
equation. Provide an economic intuition for that money demand expression.
b) Now consider the following version of the quantity theory:
M × V (i ) = P × Y
where now the velocity of money depends on the interest rate. Knowing that
the relation between the demand for real money balances and the nominal
interest rate is negative, what should be the relation between V and the
nominal interest rate? Provide an economic explanation for your result;
Solution
a)
The money demand (that is demand for real money balances M/P) can be found from
the quantity theory in the following way.
The quantity equation is:
M ×V = P × Y
From that:
M 1
= Y
P V
Or written differently:
d
M 
  = kY
 P
1
where k =
V
This money demand function is called the “Cambridge equation” or “Cambridge k
money demand” since it was first proposed by economists at Cambridge. Notice that
in that equation the focus is on money demand while in the quantity theory the focus
was on money supply. However, if the money market is in equilibrium this distinction
has no real relevance, since money supply is equal to money demand. According to
the Cambridge equation money demand is proportional to real income and the
coefficient of proportionality is the inverse of the velocity of money. If the velocity of
money is constant than also k is constant. So here, money demand is just due to
transaction motives. As real income increases, the number of transactions increases
and so does money demand.
That money demand just derived above does not explicitly take into account the role
played by the cost opportunity of holding money (the interest rate). However, it is
possible to include the cost of holding money into the Cambridge equation as we do
in part b) of this exercise.
b)
The demand function is given by:
M
= k (i )Y
P
1
. This is the same as the Cambridge equation written above with
V (i )
the difference that now the velocity of money may not be constant. This is the kind of
money demand that Keynes considered in his General Theory and it is of the type we
have seen in lecture ( L(Y , i ) ).
We know that a property of a money demand is that it is negatively related with the
∂( M / P)
cost opportunity of holding money, therefore
< 0 . Using our money demand
∂i
∂( M / P) d (k )
function we have that:
=
Y < 0 . This implies that there must be a
∂i
di
d (k )
d (V ) / di
negative relationship between k and i. Since
=−
< 0 , we must have
d (i )
[V (i)]2
dV
that
> 0.
di
where k (i ) =
That was the analytical proof of the sign of the relationship between the velocity and
the interest rate in this particular model. The intuition for that result is the following:
suppose an increase in the interest rate. Money demand decreases, so people will hold
less money now. Given the level of Y, this implies that velocity must increase. Now,
people hold less money to carry out the same amount of transactions as before (Y).
The same dollar (or pound) bill must circulate faster.
2) Suppose that the velocity of money is equal to 5, real output is fixed and equal to
10000 and that the price level is equal to 2.
a) What is the demand for real balances according to the quantity theory of
money?
b) Suppose that the government fixes the quantity of money at the level of 5000.
Determine the new price level and demand for real balances, assuming that
prices are flexible and real output and the velocity remain unchanged. What
would happen to the price level if the money supply increases to 6000?
Solution
a)
We know that the money demand is:
M
= kY
P
1
V
Given the data in the problem we have that:
where k =
M 1
= 10000
2 5
therefore the money stock M = 4000 and the demand for real balances is
M
= 2000
P
b)
Now we need to find the price level knowing the stock of money.
Using the quantity theory:
MV = PY
⇒ 5000 × 5 = 10000 × P
⇒ P = 2 .5
The demand for real balances is now:
M 5000
=
= 2000
P
2 .5
It is exactly the same as before. The reason is that the money demand here does not
depend on the nominal interest rate, but only on the REAL INCOME. Since changes
in money supply do not affect real variables, the real income stays the same, and
therefore, the demand for real balances must remain unchanged as well.
If the money supply increases till 6000, the price level will go up and again the
demand for real balances will not change.
The new price level is:
MV = PY
⇒ 6000 × 5 = 10000 × P
⇒P=3
and the new demand for real balances is still 2000.
3) (Quantity theory of money and Fisher effect). Suppose that the velocity of money V
is constant, the money supply M is growing 5% per year, real GDP Y is growing at
2% per year, and the real interest rate is r = 4%. Assume that π = π e , meaning the
ex-post inflation rate is always equal to the expected inflation rate.
a) Find the value of the nominal interest rate i in this economy;
b) If the central bank increases the money growth rate by 2% per year, find the
change in the nominal interest rate ∆i;
c) Suppose the growth rate of Y falls to 1% per year. What will happen to π ?
What must the central bank do if it wishes to keep π constant?
Solution
a)
First we need to find the inflation rate in order to obtain the nominal interest rate.
According to the quantity theory, when the velocity is constant, the inflation rate is
given by:
π=
∆M ∆Y
−
M
Y
That is the growth rate of money minus the growth rate of real income.
In this case we have that:
π = 5 − 2 = 3%
The Fisher equation says that: i = r + π e . In our case then:
i = 4 + 3 = 7%
b)
According to the quantity theory, changes in the money growth rate will translate in a
one-to-one change in the inflation rate, leaving unchanged all the real variables in the
economy (= money is neutral).
Therefore, a change of 2% in the growth rate will simply change the inflation rate by
2%, leaving the real interest rate unchanged.
Therefore, the change in the nominal interest rate is the same as the change in
inflation, therefore ∆i = 2% .
c)
According to the formula:
∆M ∆Y
−
M
Y
If the real GDP falls by 1% (so it grows a -1%) a year, while everything else is
constant, the inflation rate will increase by 1% every year.
π=
If the central bank wants to keep inflation constant, it must compensate the fall in the
real GDP by decreasing the growth rate of money supply by 1% every year.
4) Some historians have noted that during the period of gold standard (where money
was made by gold), gold discoveries were most likely to occur after a long deflation.
Provide an economic intuition for that finding.
Solution
In the gold standard all money is made by gold, so there is a fixed ratio between the
two. A deflation means a decrease in the general price level. However a decrease in
M
the general price level increases the purchasing power of money, indeed
should
P
increase (since the denominator decreases). However if the purchasing power of
money increases (the same amount of money can now buy more goods) it means that
the value of money (in terms of goods) has increased. Given that money was made
only with gold, this implies that also the value of gold increases. Therefore, after a
deflation, a given amount of gold buys more goods and services. This creates an
incentive to look for new gold deposits and, thus, more we should expect that more
gold is found after a deflation.
Notice that this is different from what happened recently about the value of gold. In
the last few years the price of gold (in dollars) has increased massively. However the
mechanism for that is different from the one in the gold standard. First we are not in a
gold standard anymore and moreover we did not see a sustained long deflation (even
if in US inflation decreased between 2009 and 2010). The reason for the recent
increase in the gold price is that it has become a better investment than many others.
Interest rates are now very low because of the crisis. This means that the returns on
many assets are now very low making those assets less attractive. Moreover gold is
considered to be fairly safe as a form of investment and therefore it has become more
popular during the recent financial crisis.

Download Report