This study develops a monetary Schumpeterian growth model with heterogeneous households and heterogeneous firms to explore the effects of inflation on innovation and income inequality. Household heterogeneity arises from an unequal distribution of wealth. Firm heterogeneity arises from random quality improvements. Under endogenous firm entry, inflation has an inverted-U effect on economic growth and income inequality. Calibrating the model for a quantitative analysis, we find that the model can match the growth-maximizing and inequality-maximizing inflation rates that are estimated using cross-country panel data. Finally, we simulate the utility-maximizing inflation rate and explore how it is affected by relative household wealth.