We examine how and why elements of organizational design depend on one another. An agent-based simulation allows us to model five features of organizations that have rarely been analyzed jointly: a vertical hierarchy that reviews proposals from subordinates, an incentive system that rewards subordinates for departmental or firm-wide performance, the decomposition of an organization's many decisions into departments, the underlying pattern of interactions among decisions, and limits on the ability of managers to process information. Interdependencies arise among these features because of a basic, general tension. To be successful, an organization must search broadly for good sets of decisions, but it must also stabilize around good decisions once discovered. Some sets of design elements encourage broad search while others promote stability. Hence, the need to balance search and stability generates interdependencies among the design elements. We pay special attention to interdependencies that involve the vertical hierarchy. We pinpoint circumstances in which a CEO who actively reviews subordinates' proposals is a bane rather than a boon, and we identify design elements that amplify or dampen the value of an active CEO. Our findings confirm many aspects of conventional wisdom about vertical hierarchies, but put boundary conditions on others.