Does the design of electoral systems entail a choice between political accountability and representation? While a significant body of research in comparative politics and political economy suggests so, others argue that these tradeoffs are contingent and weak. We develop a formal model which combines conflict between voters over policy and agency problems between voters and politicians; and we solve the model for plurality voting (PV) and proportional representation (PR) systems. At a basic level, we demonstrate that a system's representativeness and its accountability patterns are jointly determined: representing the median interests and removing low quality politicians are informed by the same set of strategic decisions. There exist both 'sweet-spots' and 'sours-pots' in the parameter space where both systems are equally effective or ineffective, respectively, in generating representation and accountability. In other contexts one system will clearly outperform the other on one or both of dimensions. The tradeoff identified in past studies, in which PV is optimal for accountability but PR optimal for representation, in fact only arises in limited circumstances; and the reverse tradeoff in which PV is optimal for representation but PR optimal for accountability is also possible. Three parameters determine the impact of electoral institutions: political polarization, social heterogeneity, and divergence between the national-level and district-level median voters. Electoral engineering becomes especially relevant in polarized electorates with imbalanced voter distributions.