Money and Banking Inventory Model of Money Demand
Transactions Demand for Money
Money is the medium of exchange, and people hold money to
make purchases.
Economists speak of the transactions demand for money, as
people demand money to make transactions.
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Money and Banking Inventory Model of Money Demand
Inventory Model of the Demand for Money
Baumol [1] presents an inventory model of the demand for
money.
A firm holds inventory so customers can buy. When the
inventory is depleted, the firm replenishes the inventory.
In the same way, an individual holds an inventory of money, to
use for purchases. When the inventory is depleted, the
individual replenishes the inventory.
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Money and Banking Inventory Model of Money Demand
Inventory Management
Consider an individual whose paycheck is automatically
deposited in a savings account at the bank, and the deposit
earns the interest rate R.
The individual periodically goes to the bank, withdraws W
dollars, gradually spends this money, and then returns to the
bank when he runs out of money.
The money holding by the individual follows a sawtooth
pattern (figure 1).
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Money and Banking Inventory Model of Money Demand
Figure 1: Money Holding
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Money and Banking Inventory Model of Money Demand
Spending
Suppose that the individual spends T dollars per year. For
example, if T = 3650, then the individual spends 10 per day.
If W = 70, then the individual would make one withdrawal per
week, 52 per year.
The total number of trips to the bank per year is
T
.
W
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Money and Banking Inventory Model of Money Demand
Cost of Trips to the Bank
Let C denote the cost of a trip to the bank. In particular, this
cost would include the value to the individual of the time taken.
The total cost per year of the trips to the bank is the cost per
trip multiplied by the number of trips,
CT
.
W
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Money and Banking Inventory Model of Money Demand
One Trip Per Year?
At one extreme, the individual might choose to make only one
trip to the bank per year, by withdrawing W = T dollars at the
beginning of the year. Throughout the year the individual
would gradually spend this money.
Of course this action would minimize the total cost of the trips
to the bank.
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Money and Banking Inventory Model of Money Demand
Loss
By making only a single trip to the bank per year, the individual
would lose interest. The dollars in the wallet potentially could
have been on deposit in the bank, earning interest.
Hence it may be better to make a smaller withdrawal but more
trips to the bank.
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Money and Banking Inventory Model of Money Demand
Foregone Interest
Since each withdrawal is W dollars, on average the individual
holds W /2 dollars in his wallet (figure 1).
The interest foregone per year is the interest rate multiplied this
average money holding,
RW
.
2
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Money and Banking Inventory Model of Money Demand
One Trip Per Day?
At the opposite extreme, the individual might go the bank daily,
to keep as much money as possible on deposit, to maximize the
interest earned.
However this action would incur a large cost for the many trips
to the bank.
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Money and Banking Inventory Model of Money Demand
Total Cost
The total cost to the individual is the cost of the trips to the
bank plus the interest foregone,
CT RW
Total Cost = + .
W 2
The optimum behavior is to choose W to minimize the total
cost.
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Money and Banking Inventory Model of Money Demand
Figure 2 graphs the two components of the total cost.
The cost of the trips to the bank follows a hyperbola, declining
as W increases.
The interest foregone follows a straight line through the origin,
as the interest foregone is proportional to W .
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Money and Banking Inventory Model of Money Demand
Figure 2: Cost
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Money and Banking Inventory Model of Money Demand
Optimum Withdrawal
The optimum withdrawal occurs when the slope of the total
cost is zero.
Setting the derivative of total cost with respect to W to zero
gives
CT RW
d(Total Cost) d W + 2 CT R
0= = =− 2 + .
dW dW W 2
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Money and Banking Inventory Model of Money Demand
By the standard calculus rule
dW n
= nW n−1 ,
dW
we have used
1
d 1
W
=− 2
dW W
dW
= 1.
dW
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Money and Banking Inventory Model of Money Demand
Square-Root Rule
Solving for the optimum withdrawal gives the square-root rule
2CT
W= .
R
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Money and Banking Inventory Model of Money Demand
Optimum Demand for Money
Since the average demand for money is W /2, the average
demand for money is
CT
M= .
2R
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Money and Banking Inventory Model of Money Demand
Higher Spending
Higher spending T raises the money demand, but less than in
proportion.
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Money and Banking Inventory Model of Money Demand
Higher Interest Rate
A higher interest rate R reduces the demand for money. To
reduce the interest foregone, the individual makes more trips to
the bank.
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Money and Banking Inventory Model of Money Demand
Higher Cost Per Trip
A higher cost per trip C increases the demand for money, as the
individual makes a larger withdrawal to reduce the number of
trips.
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Money and Banking Inventory Model of Money Demand
References
[1] William J. Baumol. The transactions demand for cash: An
inventory theoretic approach. Quarterly Journal of
Economics, LXVI(4):545–556, November 1952. HB1Q3.
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