Transitions toward sustainable food systems are urgently needed to support planetary health. While farmer and consumer behavior are key drivers of sustainability, both are constrained by entrenched power structures that currently reinforce industrialized agriculture and supply chains. These structures are rarely represented in models of agri-food systems, limiting assessments of ‘‘deep’’ leverage points for transformation. Here, we utilize two variables—capital and values—to develop a stylized agent-based model of power dynamics between farmers, consumers, markets, and the state. Simulations show that a widespread shift toward sustainability-aligned values among both farmers and consumers is necessary for a system-wide transition. Moreover, interventions to limit power concentration can enable tipping points, with transitions occurring when only 20% of farmers’ and consumers’ values shift. Our model advances understanding of how power structures interact with individuals’ behavior to produce lock-ins as well as how mobilizing sustainability-aligned values could enable more desirable futures.