The Transactions Demand for Cash: An Inventory Theoretic Approach

Baumol’s motivating analogy is that “a stock of cash is its holder’s inventory of the medium of exchange, and like an inventory of a commodity, cash is held because it can be given up at the appropriate moment” – so that a well-known result from inventory-control analysis (the economic-order-quantity or “lot size” formula, which Baumol notes had circulated since the 1920s among writers such as George F. Mellen, and had recently been analyzed by T. M. Whitin) could plausibly be applied to monetary theory (Introduction, p. 545, and fn. 1). The paper deliberately confines itself to the TRANSACTIONS demand for cash “dictated by rational behavior,” explicitly abstracting from precautionary and speculative demand by assuming transactions are perfectly foreseen and occur in a steady stream (Section I, p. 545).

An individual must pay out T dollars over a period in a steady stream; he obtains cash by withdrawing it from an interest-bearing investment (or by borrowing) in evenly spaced lots of C dollars, paying a fixed “broker’s fee” of b dollars per withdrawal, and bearing an interest opportunity cost of i dollars per dollar held per period. Since any C less than or equal to T will let him meet his payments equally well (he simply withdraws more often for smaller C), he makes T/C withdrawals per period at total brokerage cost bT/C, and since cash is spent evenly between withdrawals, his average cash holding is C/2, incurring interest cost iC/2 (Section I, pp. 545-546). The “broker’s fee” b is deliberately defined broadly to cover “opportunity losses which result from having to dispose of assets… at the moment the cash is needed,” poor resale prices from selling to a nonprofessional dealer, administrative costs, and “psychic costs (the trouble involved in making a withdrawal)” (Section I, fn. 2, p. 546).

Total cost of cash operations is bT/C + iC/2 (equation 1); setting its derivative with respect to C to zero and solving gives the paper’s central result, C = sqrt(2bT/i) (equation 2), so “the rational individual will, given the price level, demand cash in proportion to the square root of the value of his transactions” (Section I, pp. 546-547). Baumol shows in a footnote that this result is essentially unaffected if the broker’s fee has a component proportional to the amount withdrawn (b + kC), since the added term drops out upon differentiation (Section I, fn. 4, p. 547).

When receipts of T dollars occur before the corresponding stream of expenditures, Baumol allows the recipient to invest I dollars immediately and withhold R = T - I dollars as cash to cover payments until the next withdrawal from invested funds becomes necessary. Minimizing the combined cost of withholding R and investing I, together with the cost of subsequent withdrawals of size C from the invested balance, with respect to both C and I, reproduces the same square-root formula for C (with b equated to the “broker’s fee” for withdrawals) and yields R = C + T(k_w + k_d)/i for the optimal withheld balance, where k_w and k_d are the proportional components of the withdrawal and deposit fees (Section I, equations following p. 547, pp. 548-549). Since C varies with the square root of T while the second term varies in direct proportion to T, R rises less than in proportion with T – though more nearly in proportion than C does – so “the general nature of our results is… unaffected” by this extension (Section I, p. 549).

Baumol directly rebuts the view, which he attributes to several economists including Frank Knight, Divisia, and Don Patinkin, that in a stationary state there would be no demand for cash because it would always be profitable to invest all earnings and withdraw exactly the amount needed at the moment of each payment. He argues this “clearly neglects the transactions costs involved in making and collecting such loans (the ‘broker’s fee’)”; once such costs are recognized, equation (2) shows “it will generally pay to keep some cash” even in a perfectly static, fully-foreseen world, so “the analysis of a stationary monetary economy in which there is a meaningful (finite) price level does make sense” (Section II, pp. 549-550).

Baumol argues against the view – which he associates loosely with the quantity theory and explicitly with statements by Marshall, Keynes, and Pigou – that transactions demand for cash varies roughly in proportion with the money value of transactions. Since the square-root formula implies cash demand rises LESS than proportionately with transactions volume, “there are, in effect, economies of large scale in the use of cash,” and correspondingly transactions velocity (T/C) itself rises with the square root of T (Section II, pp. 550-551). He cautions, however, that “the magnitude of this difference should not be exaggerated”: equation (2) still implies that average transactions velocity varies exactly in proportion to the SQUARE of the quantity of cash, so a doubling of the cash stock, ceteris paribus, just doubles velocity (Section II, p. 551).

Baumol argues that if the square-root formula is the correct description of cash demand, and prices and interest rates are initially unresponsive to a cash injection (as might occur under widespread unemployment), then the resulting rise in transactions will be MORE than twice as large as a naive constant-velocity assumption would predict, with the excess growing with the size of the injection relative to the initial cash stock (Section II, equation following p. 551, pp. 551-552). He extends the same logic to the “Pigou effect”: since a fall in the price level raises the real purchasing power of a given nominal cash stock in a way that is “equivalent to an injection of cash with constant prices,” the same reasoning implies the stimulative effect of falling prices and wages on transactions and employment “may often have been underestimated,” which Baumol suggests may help explain the historical absence of more chronic unemployment or runaway inflation (Section II, pp. 552).

Baumol is explicit that equation (2) is, “at best,… only a suggestive oversimplification.” He lists as limitations: the rationality assumption used in the derivation; the model’s static treatment of a fixed distribution of disbursements over time (when in practice this is partly under the entrepreneur’s control); the assumption of a single, constant relevant interest rate; the assumption that the broker’s fee is constant or linear in the amount involved; the assumption of a steady stream of payments with no cash receipts during the relevant period; its restriction to a single economic unit, neglecting interactions among the cash demands of different agents in the economy; and its neglect of the precautionary and speculative demands for cash (Section III, pp. 552-553). He nonetheless argues that the model’s two basic qualitative conclusions – a positive rational transactions demand for cash, and a less-than-proportionate rise in that demand with the volume of transactions – are likely to survive relaxation of the constant-fee and constant-interest-rate assumptions under fairly general sufficient conditions worked out in an appendix-like footnote (Section III, pp. 553-554, fn. 1).

Baumol notes that if payments are “lumpy but foreseen” rather than a steady stream, cash can be used even more economically, potentially strengthening the paper’s conclusions, though imperfect foresight about lumpy payments could instead increase precautionary cash demand (Section II, point 3, pp. 554-555). On aggregation, he acknowledges that economy-wide “external economies” (cash-saving techniques spreading from one businessman to another) could reinforce individual-level savings, while “infectious liquidity fetishism” among some agents, or a rise in the “brokerage fee” itself as brokerage demand increases, could partially offset them (Section II, point 4, p. 555). On the harder problem of precautionary and speculative demand, Baumol offers little beyond a cross-reference to Arrow, Harris, and Marschak’s contemporaneous inventory-control work, but closes by noting an older parallel: Edgeworth (1888), followed by Wicksell, had already argued that if a bank’s cash demands are normally distributed, precautionary reserves held as a fixed multiple of the standard deviation of demand imply that the combined reserve requirement of several identical, independent banks rises only as the square root of their number – an early, independent instance of “square root” reasoning in banking theory that Baumol connects explicitly to his own transactions-demand result (Section II, pp. 555-556).