True or False?
The statement is false! Suppose that the interest elasticity of demand for money is infinite at interest rate levels close to zero. Then when the interest rate falls to the level at which the demand becomes perfectly elastic, money holders become willing to hold any stock of money they happen to be given, no matter how large. Monetary policy will then be ineffective because an increase in the money stock generated by an open market operation will simply add to the willingly-held inventories of money balances in the hands of money holders---they will not try to buy assets with that money because they remain in portfolio equilibrium at the higher stock of money holdings. Monetary policy is only effective when an increase in the stock of money induced by open market operations leads the private sector to re-balance portfolios by attempting to convert that money into non-monetary assets.
Fortunately, the assumption that the interest elasticity of demand for money becomes zero at a low or zero interest rates makes no sense. Nominal interest rates will not fall below zero because the real return to holding physical capital is always positive. Government bonds may have a near-zero interest rate, but it nevertheless always pays to invest in some physical capital asset---at the worst, one can buy a new car, house, or vacation property. In fact, money balances in excess of the amount needed to make exchange can be spent on luxury consumption! It is thus clear that there is a maximum money stock that would be willingly held at a nominal interest rate equal to or above zero. Monetary expansion will thus have an appropriate effect on output and employment and prices regardless of the level of nominal interest rates---an increased money stock will lead to increased aggregate spending!