Institutions are key to creating societies that are efficient, fair, and benevolent. Despite their importance, the complexities of human (networked) societies make it difficult to understand how formal institutions form and how they shape human communities. Artificial intelligence (AI) can potentially raise understanding in this regard. Thus, in this paper, we present a simulation model utilizing AI agents to simulate networked societies that contain formal institutions. We then observe the outputs of the resulting model under different societal conditions and formal institutions, and (where applicable) compare and contrast these outputs with political and economic theories. Our model outputs (a) address how inequality impacts societal prosperity, (b) illuminate how institutions can potentially impact poverty, and (c) give insights into the attributes of formal institutions that individuals are inclined to support. These and future simulation models can potentially inform how AI can support the design and development of institutions that facilitate healthier communities and nations.