Robert Leeson (Sept. 12) asks of the equation of exchange: “Does it — anywhere — address the connection between M (currency + deposits) and the monetary base (currency + reserves)?”

Indeed it does. One merely substitutes for M in the exchange equation MV = PY the identity Bm(c, r) = M relating the base B to the broad money stock M by means of the money multiplier m(c, r), an inverse function of the public’s desired cash/deposit ratio c and bankers’ desired reserves/deposit ratio r. The result is an expansion of the exchange equation to take account of the base and the money multiplier, or Bm(c,r) = PY.

In the recent financial crisis and accompanying recession, two-, three-, and even fourfold increases in the base B were largely offset by compensating falls in the multiplier m. These falls nullified or severed the transmission of stimulus from the base B to the broad money stock M. This happened partly because bankers took advantage of the Federal Reserve’s payment of positive interest on excess reserves to hold larger-than-normal amounts of those reserves — raising the reserve/deposit ratio r — rather than lending those excess reserves out in the form of newly created deposits. The rise in the reserve ratio r produced the fall in the multiplier m that partly negated the base’s B’s impact on M.