Modern advancements in energy technology, such as smart meters and renewable power generation, have contributed to the increasing penetration of time-variable electricity pricing plans. Under such plans, consumers experience a higher financial burden for consuming electricity when overall demand is high. Lower-income households may be disproportionately burdened by the transition to time-variable pricing because they tend to have less efficient homes and appliances, which may necessitate greater overall electricity consumption, especially during peak times. The goal of this work is to create a broadly applicable framework for evaluating the fairness of utility pricing plans. The proposed slope analysis method examines the distribution of a fairness metric across income levels in order to determine the level of fairness exhibited by a pricing plan. This work utilizes three fairness metrics, which are based on total household electricity bill and income, as a proof-of-concept for the slope analysis method and assesses their viability for fairness research. The proposed method also utilizes household energy models to represent various income levels for any location with sufficient data. The framework is evaluated using simulated households across 5 income levels and 3 climate zones in the United States. Fairness metrics are applied to the utility bills calculated under Real-Time Pricing and existing tariffs offered at each location. The proposed fairness evaluation method provides a quantitative measure of fairness and is broadly applicable across location and pricing plans. The metric based on the change in percentage of income spent on utilities considers the relative financial burden on households, which results in the slope analysis method outputting conclusive, accurate fairness determinations more often than the other examined metrics. The results demonstrate disparity in energy affordability, and the proposed slope analysis and model simulation methods provide a readily transferable testbed to evaluate energy policy equity.