Simple search models have equilibria where some agents accept money and others do not. We argue such equilibria should not be taken seriously - which is unfortunate if one wants a model with partial acceptability. We introduce heterogeneous agents and show partial acceptability arises naturally. There can be multiple equilibria with different degrees of acceptability. Given the type of heterogeneity we allow, the model is still simple: equilibria reduce to fixed points in [0,1]. We show that with other forms of heterogeneity, equilibria are generally fixed points in set space, and there exists no method to reduce this to a problem in R1.