We use the design-based simulation of Maples et al. (2014), which repeatedly samples from an artificial population comprised of 2008-2012 American Community Survey (ACS) unit-level data, to compare different small-area estimation models for county-level rates of school-aged children in poverty in the United States. We compare a Binomial Logit Normal (BLN) model, a Fay-Herriot model on rates, and a Fay-Herriot model on log-transformed counts inspired by the model used in the production of the official estimates by the Census Bureau’s Small Area Income and Poverty Estimates (SAIPE) Program. We also explore the effect of estimating the sampling variance on the relative performance of the models, using design-based variance estimates as well as estimates from a Generalized Variance Function (GVF). The GVF of Franco and Bell (2013, 2015) yields a considerable reduction in the Mean Squared Errors (MSEs) of the estimates of the sampling variances of the direct estimators relative to the design-based estimates, but a smaller reduction in the MSEs of the corresponding estimates of effective sample sizes. This seems to lead to an overall reduction of MSEs of the model predictors when using the GVF estimates rather than their design-based counterparts to fit the Fay Herriot model on the rates, but not so for the BLN model. Overall, the BLN model has a very modest advantage over the other two alternatives in the sense of having lower MSEs for the majority of pseudo-counties in the artificial population both when the sampling variances are known or when they are estimated.