Sticky price models featuring heterogeneous firms and systematic firm-level productivity trends deliver radically different predictions for the optimal inflation rate than their popular homogenous-firm counterparts: (i) the optimal steady-state inflation rate generically differs from zero and (ii) inflation optimally responds to productivity disturbances. We show this by aggregating a heterogeneous-firm model with sticky prices in closed form. Using firm-level data from the US Census Bureau, we estimate the historically optimal inflation path for the US economy: the optimal inflation rate ranges between 1 percent and 3 percent per year and displays a downward trend over the period 1977–2015. (JEL C51, D24, D25, E31, E52)

Investors' subjective capital gains expectations are a key element explaining stock price fluctuations. Survey measures of these expectations display excessive optimism (pessimism) at market peaks (troughs). We formally reject the hypothesis that this is compatible with rational expectations. We then incorporate subjective price beliefs with such properties into a standard asset-pricing model with rational agents (internal rationality). The model gives rise to boom-bust cycles that temporarily delink stock prices from fundamentals and quantitatively replicates many asset-pricing moments. In particular, it matches the observed strong positive correlation between the price dividend ratio and survey return expectations, which cannot be matched by rational expectations. (JEL D83, D84, G12, G14)