What are the costs of inflation fluctuations and who bears those costs? In this paper, we investigate this question by means of a quantitative incomplete-market, heterogenous-agent model wherein households hold real and nominal assets and are subject to both idiosyncratic labor income shocks and aggregate inflation risk. A key feature of our analysis is a nonhomothetic specification for households' preferences towards money and consumption goods. Unlike traditional specifications, ours allows the model to reproduce the broad features of the distribution of monetary assets (in addition to being consistent with the distribution of nonmonetary assets). Inflation risk is found to generate significant welfare losses for most households, i.e., between 1 and 1.5 percent of permanent consumption. The loss is small or even negative for households at the very top of the productivity and/or wealth distribution.