Boumal-Tobin Demand for Money, Portfolio Management

The Baumol-Tobin model is a model that explains money holdings in terms of a
transactions demand. That is, money is needed as a medium of exchange to purchase goods and
services. This note explains the algebra of optimal money demand in this model.
Jean earns Y per year, which is deposited at the beginning of the year. If Jean leaves all
the salary in money for the entire year, then Figure 1 shows that average money balances are
Y/2.
Jean realizes that this may not be a good strategy, for bonds have a higher return than
money. So Jean instead makes N trips to the bank over the course of the year. On each trip, Jean
withdraws Y/N dollars; Jean then spends the money evenly over the following 1/Nth of the
year.
How does Jean decide how many trips to make to the bank? Suppose that the cost of
going to the bank is some fixed amount F. We can view F as representing the value of the time
spent traveling to and from the bank, waiting in line to make the withdrawal, and so forth. Also,
let i denote the interest rate on bonds (or savings); because money has a zero interest rate, i
measures the opportunity cost of holding money.
If N = 1, then Figure 1 shows that average money holdings are Y/4. For any N, the
average money balances are Y/(2N), so the forgone interest is iY/(2N). Because F is the cost per
trip to the bank, the total cost of making trips to the bank is FN. The calculation then is that the
total cost is the sum of the forgone interest and the cost of trips to the bank:
Total cost = C(N) = Forgone Interest + Cost of Trips
= iY/(2N) + FN
More trips, less interest foregone and more trip cost; fewer trips, the opposite.
The optimal N, denoted N*, can be found by calculus as
dC/dN = 0 = - iY/(2N*2) + F
N* = (iY/2F)½
Optimal average money holding are
(1) M* = Y/(2N*) = (YF/2i)½
The major result here is that money demand is increasing in Y and F and decreasing in i.
Moreover, the elasticities are all one-half.
Note on inflation: We haven’t worried about prices. Assume that Y and F are real income
and cost, respectively. Further, suppose that the price of each is p. Then we can rewrite (1) with
the price level as:
(2) M*/p = Y/(2N*) = (YF/2i)½
This shows that the real money demand has an elasticity of ½ with respect to real income and
the real price of trips, and an elasticity of -½ w.r.t. the nominal interest rate.

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