The Interest-Elasticity of Transactions Demand For Cash

Tobin sets out “to support and to elaborate” Alvin Hansen’s suggestion that even the transactions demand for cash – ordinarily treated as fixed by payment habits and independent of the interest rate – becomes interest-elastic once interest rates are high enough, and to prove this rigorously rather than merely conjecture it (p. 241). The traditional account explained the ratio of cash to annual transactions purely by the synchronization of receipts and expenditures (e.g., an individual paid monthly and spending evenly through the month averages a cash balance of 1/24 of annual income), with no role for the interest rate. Tobin’s point of departure is that this account implicitly assumes transactions balances must be held entirely in cash; if transactors can instead hold part of the balance in higher-yielding bonds and shift into cash only when a payment is due, then the interest rate can affect the desired cash share even under certainty, with no other motive for holding cash than transactions needs (p. 241).

An individual receives $Y at the start of each period (t = 0) and disburses it at a uniform rate so that the balance falls linearly to zero by the end of the period (t = 1); the total transactions balance is T(t) = Y(1 - t), with average balance T-bar = Y/2 (pp. 242-243, eqs. 1-2). This balance is split at each moment between cash C(t) and bonds B(t), with B(t) + C(t) = T(t) (eq. 3). Bonds are identical to cash except that (i) they cannot be used as a medium of payment and (ii) they earn interest at rate r per period, with no default risk and no risk that r changes (p. 242). The problem is to choose the paths B(t) and C(t) that maximize interest earnings net of transaction costs.

A transaction of $x between cash and bonds, in either direction, is assumed to cost $(a + bx), where a is a fixed cost independent of transaction size and b is a cost proportional to the amount moved (p. 242). This cost is the entire reason the interest rate matters: without any cost of shifting between cash and bonds, a rational transactor would hold no cash at all except at the instant of payment, so the interest elasticity of transactions demand for cash is, in this model, a direct consequence of the cost of transacting between cash and interest-bearing assets (p. 241).

Tobin proceeds in three steps (p. 243): (1) for a fixed number of transactions n during the period, find the optimal times and amounts of those transactions, the resulting revenue R_n, and the corresponding average bond and cash holdings; (2) treating n as variable, find the revenue-maximizing number of transactions n; and (3) determine how n – and hence average bond holding B-bar and cash holding C-bar – varies with the interest rate r** (and, incidentally, with the volume of transactions Y).

Comparing alternative schedules for two transactions (Charts 1-2), Tobin derives two general principles: (a) all conversion from cash into bonds should occur at time 0, since postponing it only forgoes interest, and (b) a transaction from bonds into cash should not occur until the cash balance reaches zero, since doing it earlier only sacrifices interest that bonds would otherwise still be earning (p. 244). Applying these principles for n transactions, the optimal schedule buys (n-1)/n of Y into bonds at time 0 and sells them back into cash in n-1 equal installments at equally spaced dates, giving average bond holding B-bar_n = (n-1)/(2n) Y, revenue R_n = (n-1)/(2n) Yr, and net revenue pi_n = (n-1)/(2n) Yr - na (pp. 244-245, eqs. 5-7).

With a positive proportional cost b, a dollar’s round trip into bonds and back costs 2b regardless of speed, so it is worth buying bonds at all only if r exceeds 2b, since the maximum time a dollar can earn interest is the length of the period; bonds must be held at least 2b/r to break even (p. 244). With this modification the same equal-size, equally-spaced schedule is optimal, but with an effective starting balance of Y[1 - (2b/r)] rather than Y. The resulting general formulas are average bond holding B-bar_n = (n-1)/(2n) x Y(1 - 4b^2/r^2), revenue R_n = (n-1)/(2n) x Yr(1 - 2b/r)^2, and net revenue pi_n = R_n - na, all requiring n >= 2 and r >= 2b (p. 245, eqs. 8-10).

Comparing the constant marginal cost a of an additional transaction to the (diminishing) marginal revenue from it, Tobin identifies four possible cases depending on the size of a relative to Yr(1 - 2b/r)^2 (p. 245): (I) if a is large enough that even the second transaction is unprofitable, n = 0 and the interest rate has no effect at all on cash and bond holdings; (II) a knife-edge case where n is indeterminate between 0 and 2; (III) a case where exactly n* = 2 is optimal; and (IV) a case where n* can exceed 2 and rises as r rises.** The central conclusion follows directly: “the optimal share of bonds in a transactions balance varies directly, and the share of cash inversely, with the rate of interest” for rates falling in regimes II, III, and IV, while “within category I, of course, r can vary without affecting cash and bond holdings” (p. 246) – i.e., transactions demand for cash is interest-inelastic at low rates and interest-elastic only once rates clear a threshold set by transaction costs.

Because marginal revenue from an additional transaction rises with Y while the fixed marginal cost a does not, the optimal number of transactions n is larger, and the range of interest rates over which demand is interest-elastic (categories II-IV) is wider, the greater the volume of transactions Y* (p. 246). Concretely, “small transactors do not find it worth while even to consider holding transactions balances in assets other than cash; but large transactors may be quite sensitive to the interest rate” (p. 246), and within the interest-elastic range the ratio of cash holdings to Y falls as Y rises.

The result “suggests that the transactions velocity of money may be higher in prosperity than in depression, even if the rate of interest is constant,” since a larger volume of transactions per transactor makes economizing on cash more worthwhile (p. 246). Tobin is careful to qualify this, however: it would not be correct to conclude that aggregate transactions velocity depends directly on the level of money income, because what matters is the volume of transactions Y relative to the transaction cost a, and “in a pure price inflation Y and a could be expected to rise in the same proportion” – so the mechanism does not straightforwardly predict that velocity rises with the price level (p. 246).

Tobin acknowledges in a footnote that he had not read Baumol’s Quarterly Journal of Economics paper before writing his own, and the Appendix compares the two: the maximization underlying Tobin’s equation (10) yields “essentially the same result” as Baumol’s revenue expression, but with three differences (pp. 246-247). First, Tobin treats the number of transactions n-1 as restricted to positive integers, while Baumol treats the analogous variable as continuous. Second, Tobin proves what Baumol simply assumes – that optimal cash withdrawals should be equally spaced in time and equal in size. Third, Baumol does not consider the possibility that the optimal initial investment in bonds is zero (Tobin’s Regime I), partly because Baumol frames the problem as minimizing an interest cost charged on the average cash balance rather than maximizing net interest earnings – a framing Tobin criticizes as resting on an implicit zero-cash benchmark that would itself require infinitely many transactions and therefore infinite transaction costs (p. 247). This paper and Baumol’s are the twin sources of what later became known as the Baumol-Tobin model of money demand.