We analyze the merits and limitations of simple tractable New Keynesian models (RANK [representative agent New Keynesian] and TANK [two-agent New Keynesian]) in accounting for the aggregate predictions of heterogeneous agent New Keynesian models (HANK). By means of comparison of a number of nested HANK models, we investigate the role played by (1) idiosyncratic income risk, (2) a binding borrowing constraint, and (3) a portfolio choice between liquid and illiquid assets. We argue that the effects of household heterogeneity can be largely understood by looking at the differential behavior of two types of households: hand-to-mouth and unconstrained. We find that a suitably specified and calibrated TANK model (which abstracts from idiosyncratic income risk) can capture reasonably well the aggregate implications of household heterogeneity and the main channels through which it operates. That ability increases in the presence of a policy rule that emphasizes inflation stability. In the limiting case of a strict inflation-targeting policy, heterogeneity becomes irrelevant for the determination of aggregate output.