Two special cases for the slope of the LM curve deserves to be mentioned. One when the interest elasticity of the Md is zero and the other when it is extremely high.
When the interest elasticity of the Md is zero i.e. h = 0. When Y increases in this case there is no possible rise in interest rate that can reduce the demand for money back to the level of the fixed
money supply. Reason being that a rise in interest rate cannot cause people to reduce their speculative demand for money or to economize on transactions balances. Consequently, the money market can be at equilibrium at only one level of income. Therefore the LM curve is this case is vertical.
The other extreme case occurs when the interest elasticity of money demand becomes extremely large, approaching infinity. This occurs when there is a liquidity trap i.e. a situation when interest rate is so low relative to what is considered normal, that there is now a general consensus that interest rate will rise. In this situation, expected future losses will outweigh the interest earnings on bonds. The public would rather hold any increase in money balances (with negligible fall in interest rate). When we have the liquidity trap case, the LM curve is horizontal.
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