## Monitoring multiple agents: the role of hierarchies |

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action agency relations agency theory agent's wage aggregate output analysis analyzed arguments assume Binomial distribution characterized concave concave function consider constraints convergence convex function convexity cost of employing define derive dichotomous distribution of output equations example Gjesdal hence hierarchical organization identical implicit function theorem implies incentive integer Jensen's inequality lower level supervisors marginal value Markov matrix MAX1 monitoring effort monitoring hierarchy monitoring technologies multi-agent setting multiple agent Nash equilibrium notation number of agents number of supervisors number of workers observed optimal allocation optimal contract optimal hierarchy optimal span order condition output-dependent contract paper parameters principal principal-agent problem probability production profit PROOF properties Proposition reservation utility result returns to scale risk neutral satisfies section 3.1 shirking smooth function solved span of control stochastic strictly increasing supervisor with span supervisor's wage theorem top-level supervisor utility functions wage contracts wage cost wage function zero