The Square-root Formula in Monetary Theory

It is well known that in the classical monetary theory, whether of the Fisherian "equation of exchange” version or the Cambridge real-balance version, the demand for money was seen strictly as a demand for transactions purposes, which in turn was exogenously determined by institutional and technological factors.1 One of the innovations which Keynes introduced into monetary theory was his explicit recognition that the demand for money depends not only on income, but also on the yield on some alternative financial asset, typically a government bond. Keynes [42] dichotomized the demand for money into two parts: the first part, the transactions and precautionary demand depends, as in the classical model, on income, while the other part, the speculative demand, was treated as a function of the market rate of interest. Subsequently in the fifties, some economists began to argue that even the transactions demand for money is sensitive to the rate of interest, and that there are important economies of scale in the holding of cash for transactions purposes. This hypothesis, crystallized in the famous square-root formula, has dominated monetary thinking since the mid-fifties. Indeed, in some writings it has been referred to as the Square-Root Law. In recent years, however, this formulation of the transactions demand has been subjected to a critical re-examination from both the theoretical and empirical points of view. The relevant literature has grown to such an extent that a brief survey is called for. The purpose of this paper is to re-appraise the inventory theoretic approach in general and the square-root formula in particular, in the light of recent theoretical discussions and empirical tests. Section I summarizes the formulation of the squareroot hypothesis and its extensions. Section II reviews recent critiques and tests of the hypothesis. Section III then contains some concluding remarks.