Excitatory and inhibitory inputs to cortical neurons remain balanced across different conditions. The balanced network model (Van Vreeswijk & Sompolinsky 1998) provides a self-consistent account of this observation: under mild inequality conditions on connectivity parameters, population rates dynamically adjust to yield a state in which inputs to all neurons are tightly balanced, keeping all populations active at biological levels. Global tight balance, however, predicts a linear stimulus-dependence for population responses and cannot explain systematic cortical response nonlinearities such as divisive normalization, a form of sublinear integration and a canonical brain computation (Carandini & Heeger 2012). Nevertheless, when necessary connectivity conditions for global balance fail, states arise in which only a subset of neurons are active and have balanced inputs. While such localized balance states open the door to nonlinear behavior (Baker et al. 2020), it is unknown if they yield sublinear integration. Here we show that in networks of neurons with different stimulus preferences, localized balance robustly leads to sublinear integration, including normalization and winner-take-all behavior. We analytically quantify these effects and derive inequality conditions for their emergence. An alternative model that exhibits normalization is the stabilized supralinear network (SSN), which predicts a regime of loose, rather than tight, excitatory-inhibitory balance (Ahmadian & Miller 2021). However, an understanding of the causal relationship between excitatory-inhibitory balance and normalization in SSN and conditions under which SSN yields significant sublinear integration are lacking. For weak inputs, SSN integrates inputs supralinearly, while for very strong inputs it approaches a regime of tight balance. We show that when this latter regime is globally balanced, SSN cannot exhibit strong normalization for any input strength; thus, in SSN too, significant normalization requires localized balance. In summary, we causally and quantitatively connect a fundamental feature of the dynamics of cortical excitation and inhibition with a canonical brain computation.