Comparisons of Experiments in Moral Hazard Problems

TL;DR

This paper introduces a geometric framework to compare information structures in moral hazard problems, establishing multiple equivalent characterizations and applying them to various incentive and cost comparison scenarios.

Contribution

It develops a novel geometric approach with three nested orders to analyze and compare information in moral hazard problems, linking geometric, algebraic, and economic characterizations.

Findings

Column space order characterizes implementability comparisons.

Conic span order relates to cost comparisons with risk-neutral agents.

Zonotope order applies to cost comparisons with risk-averse agents.

Abstract

I use a novel geometric approach to compare information in moral hazard problems. I study three nested geometric orders on information, namely the column space, the conic span, and the zonotope orders. The orders are defined by the inclusion of the column space, the conic span, and the zonotope of the matrices representing the experiments. For each order, I establish four equivalent characterizations of the orders, (i) inclusion of feasible state-dependent utility sets, (ii) matrix factorizations, (iii) posterior belief distributions, and (iv) improved incentives in certain moral hazard problems. The column space order characterizes the comparisons of implementability in all moral hazard problems. The conic span order characterizes the comparisons of costs in all moral hazard problems with a risk neutral agent and limited liability. The zonotope order characterizes the comparisons of costs in all moral hazard problems when the agent can have any utility exhibiting risk aversion.