We investigate in details a Trejos-Wright random matching model of money with a consumer take-it-or-leave-it o¤er and the individual money holding set f0; 1; 2g. First we show generic existence of three kinds of steady states: (1) pure-strategy full-support steady states, (2) mixed-strategy full-support steady states, and (3) non-full-support steady states, and then we show relations between them. Finally we provide stability analyses. It is shown that (1) and (2) are locally stable, (1) being also determinate. (3) is shown to be unstable. (JEL classi…cation: C62, C78, E40)