Many electricity demand estimates have been obtained based on the assumption that consumers optimize with respect to known marginal prices, but increasing empirical evidence suggests that consumers are more likely to respond to average prices. Under this assumption, this paper develops a new strategy based on Generalized Method of Moments to estimate household electricity demand. Our demand estimation approach uses publicly available expenditure data and utility-level consumption data from several major U.S. cities, complementing studies that use individual billing data which are richer yet often proprietary. We estimate the price elasticity near - 0.50, which is at the upper end (in magnitude) among the estimates from previous studies. This could have important implications for policy analysis such as those on climate policies that may affect electricity prices.