The Transactions Demand for Cash: An Inventory Theoretic Approach

This 1952 Quarterly Journal of Economics paper by William J. Baumol applies inventory-control theory to the transactions demand for cash, reasoning that “a stock of cash is its holder’s inventory of the medium of exchange” and that “inventory theory and monetary theory can learn from one another” (Introduction, p. 545). In the simple model of Section I, an individual pays out T dollars in a steady stream over a period, obtaining cash by withdrawing it from an interest-earning investment (or by borrowing) in evenly spaced lots of C dollars, incurring a fixed “broker’s fee” b per withdrawal (a deliberately broad category covering all non-interest costs of obtaining cash) plus an interest opportunity cost of i dollars per dollar held per period; minimizing the sum of brokerage and interest costs with respect to C yields the paper’s central “square root formula,” C = sqrt(2bT/i), so the optimal cash withdrawal – and hence the average cash balance C/2 – rises only with the square root of the volume of transactions T, not in direct proportion to it (Section I, pp. 545-547). Baumol extends this to the case where cash receipts precede expenditures, showing that the recipient’s optimal choice of how much to invest immediately versus withhold as cash again yields the same square-root relationship for the working cash balance, with the withheld balance R rising less than proportionately with T (though more nearly in proportion than C) (Section I, pp. 547-549). In Section II, Baumol draws out several consequences of the square-root formula: it refutes the claim that a stationary economy would exhibit no demand for cash at all, since positive cash holding remains cost-minimizing even in a static setting once transactions costs are recognized (pp. 549-550); it implies that the transactions demand for cash rises less than in proportion with the value of transactions, producing built-in “economies of large scale in the use of cash” and hence a rising transactions velocity of money as transaction volumes grow (pp. 550-551); and it suggests that the effect of a cash injection on transactions and employment, and correspondingly the strength of the Pigou (real-balance) effect from falling prices, may have been systematically underestimated by economists who assumed a constant, proportional relationship between cash and transactions (pp. 551-552). Section III candidly catalogs the model’s limitations – its assumption of a single constant interest rate, its treatment of the “broker’s fee” as constant or linear, its restriction to a single economic unit with a perfectly foreseen, steady stream of payments, and its neglect of precautionary and speculative cash demands – while arguing that the paper’s two basic qualitative conclusions (a positive, less-than-proportionate transactions demand for cash) are likely to survive relaxation of these assumptions under fairly general conditions (Section III, pp. 552-554), and closes by noting an older, independently derived “square root” result in banking theory (Edgeworth 1888, following by Wicksell) for precautionary reserves, which Baumol connects to his own analysis (Section II, p. 556).

William J. Baumol | Macro Paper Warehouse

The Transactions Demand for Cash: An Inventory Theoretic Approach