Dynamic Hierarchies and Organizational Capital: A Unified Model of Incentives, Structure, and Inequality

This paper develops a continuous-time model of multi-layer hierarchies that integrates classic theories of firm organization with recent advances in organizational economics. Building on the branching frame work of Williamson (1967), Calvo and Wellisz (1978), Keren and Levhari (1979-83), and Mirrlees (1976), and incorporating modern concepts of organizational capital (Dessein and Prat 2022; Bloom, Sadun, and Van Reenen 2021), we endogenize effort, wages, monitoring, managerial at- tention, span of control, and hierarchical depth. A representative owner maximizes the present value of profits by choosing continuous profles of these controls and investment in organizational capital. We derive a full Hamiltonian system of first-order conditions, show existence of balanced growth steady states, and analyze the long-run dynamics of firm size and wage inequality. The model predicts that higher productivity or cheaper monitoring induces more effort, wider spans, deeper hierarchies, and faster organizational-capital accumulation, while adverse shocks have the oppo site effects. Internal wages decline with hierarchical depth and exhibit rising dispersion when productivity growth outpaces monitoring improve ments, offering a micro-foundation for recent empirical findings on firm growth and inequality. Our framework nests classical discrete hierarchies as a special case and provides a versatile platform for empirical calibration, stochastic extensions, and general-equilibrium applications.