Search models of monetary exchange have typically relied on Nash (1950) bargaining, or strategic games that yield an equivalent outcome, to determine the terms of trade. By considering alternative axiomatic bargaining solutions in a simple search model with divisible money, we show that the properties of the bargaining solutions do matter both qualitatively and quantitatively for questions of first importance in monetary economics such as: (i) the efficiency of monetary equilibrium; (ii) the optimality of the Friedman rule; (iii) the welfare cost of inflation