Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Incentive bonus and limited liability rent for unobserved effort paid on an imperfect indicator

h(c, p_H, p_L, ell) = (b_star, E_pay, R)

Maps the extra cost to an agent of a high effort level, the probabilities that a verifiable indicator is met with and without that effort, and the most the agent can be made to pay the principal, to the smallest bonus per indicator met that makes the agent choose the high effort, the principal's expected payment and the agent's expected rent. This is the hidden action, or moral hazard, form of the principal-agent problem with a risk-neutral agent protected by limited liability, as in chapter 4 of Laffont and Martimort. The mechanism-design records HE-FN-MECH-001, HE-FM-MECH-001, HE-FM-MECH-002 and HE-CF-MECH-001 treat hidden information instead, a supplier's cost type known only to itself and screened with a menu of contracts, and the Ellis and McGuire records HE-FN-CTH-001 to HE-CF-CTH-001 set a mixed payment on the case and its recorded cost, both verifiable, with no payment on an indicator of unobserved effort. Demand-side moral hazard in insurance is HE-FM-MH-001 and HE-FM-MHHI-002. The expected payment is a two-branch expected value of the form of HE-FM-CHN-001, and the rent takes the form of the margin of HE-FM-CTH-001 with no reimbursed share. Whether the principal should buy the effort at all compares the gain in its expected benefit with the expected payment, condition (4.23) of Laffont and Martimort when ell = 0, a single comparison that needs no separate formula (HE-ER-PAP-002). Multitasking and gaming are described in the article without an equation. Notation follows the Principal-Agent Problem article, with the liability limit ell taken from Laffont and Martimort.

  • Smallest effort bonus, expected payment and limited liability rent for a binary effort choice

    b_star = c / (p_H - p_L); F = min(ell, p_L * b_star); E_pay = p_H * b_star - F; R = E_pay - c

    A risk-neutral agent, such as a general practice, chooses a high effort that costs it c per patient or a low effort that costs nothing, and is paid a bonus for each patient whose recorded indicator is met, which happens with probability p_H after high effort and p_L otherwise. The smallest bonus that satisfies the incentive constraint, p_H b minus c at least p_L b, is b_star = c / (p_H minus p_L), the gap between the transfers (4.20) and (4.19) of Laffont and Martimort, and the principal also charges a fixed fee F per patient, the smaller of the liability limit ell and p_L b_star, which combines the two cases of their Proposition 4.2 (derived). The expected payment, the two-branch expected value of HE-FM-CHN-001 with branches b_star minus F and minus F, is the second-best cost of inducing effort, and the rent R is that payment less c, the form of the HE-FM-CTH-001 margin with no reimbursed share. With ell = 0 the rent is c p_L / (p_H minus p_L), which equals what the agent would earn without the effort and, for p_L above 0, c / (p_H / p_L minus 1) (derived), the limited liability rent (4.21) that the authors describe as the extra payment caused by moral hazard combined with limited liability. At ell equal to p_L b_star the rent (4.21) of the second case of their Proposition 4.2 falls to zero, and for a larger ell the first case applies, with no rent.