Class7_Money. Detailed information on money

Money 2: Money Market
Class 7
Wonmun Shin
(wonmun.shin@sejong.ac.kr)
Department of Economics, Sejong University
* This lecture note is written based on Professor Xavier Sala-i-Martin’s lecture notes.
Macroeconomics (Fall 2021) Money (2) 1 / 25

Money Demand: Baumol-Tobin Model

Demand for Money Balance
Recall, the DBC when we introduce money:
PtCt + Bt + Mt
| {z }
Uses of Income
= PtYt + (1 + i) Bt−1 + Mt−1
| {z }
Sources of Income
Note that money balance (or money stock) Mt is one of the uses of income.
In other words, an individual allocates some resources to consumption, some
to bond holdings, and some to money itself.
Here, we can see that money is another type of financial asset.
So far, bond (B) was the only financial instrument. Now, we allow individuals
to choose between two financial assets which are bond holding (B) and
money balance (M).
That is, you can choose between saving in the bank (that yields interest
earnings) and cash in your pocket (that yields no earning).

Demand for Money Balance [cont’d]
The opportunity cost of holding money is being able to save that money in
bonds at the nominal interest rate instead.
Money is the financial asset for purchasing goods, and on the other hand,
bonds are the asset for yielding interest.
Hence, there is a trade-off between holding money and bonds.
If you hold all your wealth in your pocket as cash, you give up huge interest!
On the other hand, if you have all your wealth in the bank, you don’t lose any
interest. But whenever you need to purchase goods, you need to go back to the
bank, change bonds for money and then run to the store and finally purchase
goods you wanted to get.
There is a cost associated to changing bonds for money!
Due to the cost, people wants to hold some money in their pocket.
Then, what is the optimal amount of money balance when there is a trade-off
between holding money and bonds?

Baumol-Tobin Model
The optimality question is therefore, how much money an individual wants to
hold on average, given the trade-off.
Assume that an individual wants to spend PY dollars during a period of
length T (say, a month).
In other words, we are assuming that this individual receives an income of PY
dollars every month and has decided to spend it fully by the end of the month.
Assume that the individual saves all PY income in the bank as soon as he
receives it at the beginning of the month.
Now, this individual simply needs to decide how much of the income he
wants to withdraw and carry every day in his pockets.
However, we would not be interested in knowing how much money this
individual actually carries in day 1, in day 2, and so on.
Rather, we are interested in how much money he carries on average over the
30 days of the month.
Let us call the daily average demand of money Md .

Baumol-Tobin Model [cont’d]
Also assume that the individual smoothes consumption over the month.
That is, he spends the amount of money in his pocket at a steady rate
because he uses the same amount of cash every day to buy goods.
Let us define N as the number of times the individual goes to the banks to
withdraw some portion of his PY income.
Intuitively, there should be a negative relation between N and Md .
The less money we want to hold at all points in time (lower Md ), the more
times we will have to go to the bank (higher N).
If the money we have on our hands is little, then it goes away quickly as we
spend it in consumption at a steady rate and so we have to go more often to
the bank to get more cash.
On the other hand, if we want to go to the bank very few times, we will have
to carry lots of cash every day and only use a small fixed amount of money to
consume daily.

Let’s see how the desired average holding of money decreases if we increase the number of
trips to the bank.
N = 1: we withdraw PY money as soon as we save all income in the bank.
Since our income is depleted at the steady rate by which we we smoothly consume,
our PY money declines steadily to zero from time 0 to T.
Note that the accumulated money holdings between time 0 to T is the area of 4ABC.

So, the average daily money holdings Md is the height of DBCF, which is PY /2.
If Md = PY
2 , then AD = FC in length. And AC and DF are straight lines intersecting
each other with common (→ ]AED = ]CEF) . Also, ]ADE = ]CFE).
Therefore, 4ADE and 4ECF are congruent and thus share the same area.
As a result, the area of 4ABC and the area of DBCF are the same when the
height of DBCF is PY /2.

This means Md = PY
2 is the average money holdings throughout the period
T.
Hence, though we only made one trip to the bank (N = 1) and we were
holding a lot of money in our pockets during the initial days and very little
towards the end, this journey is equivalent to have been carrying PY /2
dollars in our pockets every day steadily.
As a result:
Md
=
PY
2
when N = 1

Repeating the procedure but now going to the bank twice, i.e. N = 2, we get
Md = PY /4

For N = 4, Md = PY /8.

In general, if we go to the bank N times, then:
Md
=
PY
2N
This confirms the intuition we had anticipated earlier: The more times we go
to the bank, the less money we need to get each time we go.
Solving for N:
N =
PY
2Md
(1)

Now that we established the relationship between nominal demand for money
and number of trips to the bank, we can use this relationship to make explicit
what is the total cost of holding money in terms of Md !
Total Cost of Holding Money (TC) = Financial Cost + Transaction Cost
Financial cost of holding money: nominal interest rate earnings foregone for
holding money instead of putting it in the bank (or buying bonds)
Financial Cost = i × Md
Transaction cost of holding money: the amount of money you need to pay
to go to the bank N times
Transaction Cost = N × P · ψ
where ψ is the real transaction cost (and so P · ψ is the nominal transaction
cost).

TC = iMd
+ NPψ (2)
Substituting the value of N from the equation (1) into (2):
TC = iMd
+

PY
2Md

Pψ
= iMd
+
ψP2Y
2Md
To find the optimal level of Md , we want to find the size of Md that
minimizes TC.
dTC
dMd
= i −
ψP2Y
2 (Md )
2
= 0

Md
P
=
r
ψY
2i
Md
P is the real money demand.
The real money demand is decreasing in i.
The higher the nominal interest rate i (all else being equal), the higher the
losses of holding real money balance.
This is because we can deposit our money in the bank and get a higher return
than before i went up.
Thus, Md
P goes down as i goes up.

Md
P
=
r
ψY
2i
The real money demand is increasing in Y .
The higher our income is, the more we want to consume because we have
more resources available, so the more money we need.
Thus, Md
P goes up as Y goes up.
The real money demand is increasing in ψ.
The higher the transaction cost of getting money, the more expensive it is to
go to bank (so we go less).
Since we avoid going to the bank, we will bear the burden of having to carry
more money in our pockets, which means we demand more money.
Thus, Md
P goes up as ψ goes up.

Demand for Real Money Balance
Let us summarize the results of Baumol-Tobin model by writing the
following expression for real money demand:
Md
P
= L (i, Y , ψ)
Note that Li 0, LY 0, and Lψ 0.
Question: Why do we find an expression for real money demand Md /P,
instead of simply finding an expression for nominal money demand Md ?
Answer: What we care about is real money balance, which is the purchasing
power of Md dollars.
In other words, what it is relevant to individuals is not how much they get paid
nominally but how many things they can purchase.
Therefore, we need to know how much real balances we demand.

Money Demand Function
Finally, we can obtain the linear relationship between Md and P:
Md
= P · L (i, Y , ψ) = P
r
ψY
2i
!

Money Demand Function [cont’d]
Y ↑ → Higher demand for real balances → For every level of prices (P), Md is higher (that
is, Md curve shifts right).
i ↑ → Higher opportunity cost of holding money → For every level of P, Md is lower (that
is, Md curve shifts left).
Changes in P cause a movement along the curve.

Money Supply
Now that we finally derived the demand for money, we are only in need of
one more ingredient to find the equilibrium in the money market. → Money
supply!
Assumption: Money supply is exogenous.
Central bank (or government) decides the money supply Ms.
It sets Ms exogenously to the model at a constant value.
Ms
= M
Since money supply does not depend on P, the money supply curve is
completely vertical at M.

Money Supply [cont’d]
Ms is not affected by the changes in P.
M ↑ → Shift of line to the right. // M ↓ → Shift of line to the left.

Equilibrium in Money Market
With the demand and supply for money at hand, now we can find the equilibrium in the
money market.
The equilibrium takes place when the quantity demanded of money is equal to the quantity
supplied.
In the figure below, this equilibrium happens at P∗ where the two curves intersect.
That is, at P∗, money demand equals the exogenous money supply.

Equilibrium in Money Market [cont’d]
Let us see how the equilibrium dynamics work.
Suppose the economy was at low price level: Pl P∗ → Excess money
supply Ms Md
Long line of consumers will be outside of the stores with pockets full of money,
to buy goods.
The owners of the stores will then realize they can afford to raise the price
level because a lot of people would still be outside their doors.
This process continues until P = P∗.
Conversely, when the economy is at high price, Ph P∗, there is scarcity of
monetary assets.
Given this scarcity of money, the owners of the stores choose to lower prices to
attract more consumers.
This process continues until P = P∗.

(optional) Quantity Theory of Money
The above dynamics of equilibrium is a result of the Quantity Theory of Money (QTM).
Transaction velocity of money (V ): the velocity at which money is spent during a given
short-term period (or, the average frequency across all transactions with which a unit of
money is spent)
V =
PY
M
QTM assumes that V is constant in the short run.
That is, if Y is constant (∵ short run), then M and P move in the same direction.
QTM is usually expressed in the following form:
MV = PY
Classical (specifically, Monetarist): QTM predicts that, if the amount of money in
an economy double, then price level will also double.
Keynesian: V is not stable, and P is sticky in the short run, so the direct relationship
between money supply and price level does not hold.
Anyway, our Baumol-Tobin model is a more general version of QTM given that it
introduces the role of interest rates into the story besides income.
The reason we bring QTM is to help us with the intuition of the equilibrium dynamics.

Increase in Money Supply
Rise in money supply → Ms curve shifts to the right.
At the initial P∗
1 , there is an excess of money supply.
The owners of stores increase prices when they see a long line of people waiting
outside the stores.
Higher prices imply higher demand for money (Md = P · L (i, Y , ψ)).
Individuals need more money to keep their purchasing power at the same level.
As a result, prices and money demand would increase simultaneously along the curve (to
equal money supply): A → B

Increase in Nominal Interest Rate
Rise in i → Md curve shifts to the left.
At the initial P∗
1 , there is an excess of money supply.
Prices and money demand would increase simultaneously along the new Md curve (to
equal unchanged money supply): A → B
As a result, price increases (P∗
1 → P∗
2 ) but there is no change in the quantity of money
(M∗) because money supply was held constant.

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