Signature
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
| Inputs | Definition | Unit |
|---|---|---|
c | Cost to the agent of the high effort, such as the staff time of a thorough review, over the low effort, which costs nothing; psi in Laffont and Martimort | pounds per patient, above 0 |
p_H | Probability that the recorded indicator is met for a patient when the agent makes the high effort; pi_1 in Laffont and Martimort | probability, above p_L and at most 1 |
p_L | Probability that the recorded indicator is met for a patient when the agent makes the low effort; pi_0 in Laffont and Martimort | probability, at least 0 and below p_H |
ell | Most the agent can be made to pay the principal per patient, such as a fixed fee charged whatever the indicator shows; 0 when the agent cannot be paid less than nothing | pounds per patient, at least 0 |
b_star | Smallest bonus per patient whose indicator is met that makes high effort at least as good for the agent as low effort, the gap between the payments when the indicator is met and when it is not | pounds per patient meeting the indicator |
|---|---|---|
F | Fee the principal charges per patient, the smaller of ell and p_L b_star, so that the agent receives b_star minus F when the indicator is met and pays F when it is not | pounds per patient, at least 0 |
E_pay | Expected payment per patient to an agent that makes the high effort, p_H b_star minus F, the second-best cost of inducing effort | pounds per patient |
R | Expected payment less the cost of the high effort, the agent's limited liability rent and the principal's cost of inducing effort above c | pounds per patient, at least 0 |
Function
Incentive bonus and limited liability rent for unobserved effort paid on an imperfect indicator
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.
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Implementations
Excel
Excel bonus, fee, expected payment and rent for a binary effort choice
With the cost of the high effort in a cell named ReviewCost, the two probabilities in ProbMetThorough and ProbMetCursory and the liability limit in LiabilityLimit, the four formulas return the bonus in a cell named BonusPerIndicator, the fee in FeeCharged, the expected payment in ExpectedPayment and the rent in PracticeRent. N() turns a blank LiabilityLimit into 0; without it MIN would ignore the blank cell, return ProbMetCursory times BonusPerIndicator as the fee and report no rent. Equal probabilities return #DIV/0! in every cell.
=ReviewCost/(ProbMetThorough-ProbMetCursory); =MIN(N(LiabilityLimit),ProbMetCursory*BonusPerIndicator); =ProbMetThorough*BonusPerIndicator-FeeCharged; =ExpectedPayment-ReviewCost
Assumptions
Two effort levels and an indicator linked to effort only through its probability
The agent chooses between a high and a low effort, the principal sees only whether a verifiable indicator is met, and the indicator is met with probability p_H after high effort and p_L after low effort, with p_H above p_L, as in section 4.2 of Laffont and Martimort. The payment can depend on the indicator and on nothing else about the agent's work. With p_H equal to p_L the indicator says nothing about effort, no bonus induces the high effort and the formula divides by zero; with p_H below p_L it returns a negative bonus and does not apply (derived).
Risk-neutral parties and a floor on payments to the agent
Both parties are risk neutral and a payment to the agent can fall no lower than minus ell, the limited liability constraints (4.17) and (4.18) of Laffont and Martimort. The practice in the article cannot be paid less than nothing, so ell = 0. With a risk-averse agent the trade-off is instead between insurance and efficiency, section 4.5 of the same text, which this formula does not cover. The agent's outside option is zero, and at b_star its expected payoff from the high effort is R, never below zero, so it accepts the contract (derived).
Weak incentive constraint and an indifferent agent at the smallest bonus
The incentive constraint, p_H b minus c at least p_L b, is a weak inequality, as (4.3) and (4.15) of Laffont and Martimort are, and their Definition 4.1 calls a contract that satisfies (4.3) and (4.4), with equality allowed, incentive feasible, which the text defines as inducing a positive effort. At b_star the agent is indifferent and is taken to make the high effort. Remark 2 of the same chapter notes that other transfers can give the agent a strict preference for effort, so a principal that wants a margin against the tie pays slightly more than b_star.
One task and an indicator the agent cannot move without the effort
The agent has one task and can change the chance that the indicator is met only through its effort. The article describes multitasking, where a bonus on a measured task draws effort away from unmeasured ones, and gaming such as exception reporting, which raises the recorded indicator without the review. Gaming of this kind raises p_L and with it the rent, since R rises with p_L for fixed c and p_H (derived).
Worked examples
Weak quality indicator for a diabetes annual review
The article's illustrative example. A purchaser wants a general practice to carry out a thorough annual review for each patient with diabetes, costing the practice 30 pounds of staff time, and can pay only on an indicator met for 70% of patients after a thorough review and 45% without one. The practice cannot be paid less than nothing. The bonus is 120 pounds per patient meeting the indicator, the expected payment 84 pounds and the rent 54 pounds per patient, what the practice would earn by chance without any review. For a list of 200 patients the purchaser expects to pay 16,800 pounds for reviews costing 6,000 pounds, a rent of 10,800 pounds.
c = 30; p_H = 0.70; p_L = 0.45; ell = 0; b_star = 120; F = 0; E_pay = 84; R = 54
Sharper audited indicator for the diabetes annual review
The article's illustrative second case. An audited measure is met for 90% of patients after a thorough review and 20% without one, and the practice still cannot be paid less than nothing. The bonus falls to 42.86 pounds, the expected payment to 38.57 pounds and the rent to 8.57 pounds per patient, or 1,714 pounds for a list of 200. The weak indicator of HE-EX-PAP-001 costs the purchaser about six times as much in rent, because p_H / p_L is 4.5 rather than 1.56.
c = 30; p_H = 0.90; p_L = 0.20; ell = 0; b_star = 42.86; F = 0; E_pay = 38.57; R = 8.57
Fixed fee of 54 pounds removes the practice rent
The article's illustrative aside. If the purchaser could also charge the practice a fixed fee of 54 pounds per patient, the weak indicator and the same 120 pound bonus still buy the review. The practice receives 66 pounds when the indicator is met and pays 54 pounds when it is not, the transfers (4.12) and (4.13) of Laffont and Martimort, which at this liability limit coincide with (4.19) and (4.20), so the expected payment equals the cost of the review and the rent is zero.
c = 30; p_H = 0.70; p_L = 0.45; ell = 54; b_star = 120; F = 54; E_pay = 30; R = 0
Practice able to pay only 20 pounds per patient
Illustrative inputs. With the weak indicator and a practice that can be charged at most 20 pounds per patient, the fee is 20 pounds, the expected payment 64 pounds and the rent 34 pounds, the 54 pounds of HE-EX-PAP-001 less the fee, as (4.21) of Laffont and Martimort gives for a liability limit below p_L b_star.
c = 30; p_H = 0.70; p_L = 0.45; ell = 20; b_star = 120; F = 20; E_pay = 64; R = 34
Limiting case of an indicator never met without the review
Illustrative inputs. When the indicator is met for 90% of patients after a thorough review and never without one, the bonus is 33.33 pounds, the expected payment equals the 30 pound cost and the rent is zero even though the practice cannot be charged a fee, because nothing is paid for indicators met by chance. The case checks an implementation.
c = 30; p_H = 0.90; p_L = 0; ell = 0; b_star = 33.33; F = 0; E_pay = 30; R = 0
Common errors
Bonus set so that the expected payment just covers the cost
Setting the bonus to c / p_H, so that a practice making the review is paid its cost on average, ignores that the indicator is also met without the review. With the weak indicator of HE-EX-PAP-001 the bonus is 42.86 pounds, the practice earns nothing above cost from the thorough review and 19.29 pounds per patient without it, so it skips the review and the purchaser pays 19.29 pounds per patient for indicators met by chance (derived).
Judging whether to buy the effort against its cost alone
For an agent that cannot be charged a fee, condition (4.23) of Laffont and Martimort induces effort only when the gain in expected benefit is at least the second-best cost, E_pay here, which exceeds the first-best cost c by the rent, so with limited liability some effort worth more than its cost is not worth buying (their Figure 4.3 and Proposition 4.3). With the weak indicator of HE-EX-PAP-001 a review worth at least 30 and less than 84 pounds per patient to the purchaser would be bought if effort were observed but not through the bonus; with the sharper indicator of HE-EX-PAP-002 that band narrows to at least 30 and less than about 38.57 pounds (derived).
Probabilities typed as percentages
Entering p_H and p_L as 70 and 45 makes the difference 25 and the bonus 1.20 pounds. The practice's real chances are still 0.70 and 0.45, so the review raises its expected bonus by 0.30 pounds against a cost of 30 pounds, it skips the review, and the purchaser pays 0.54 pounds per patient for indicators met by chance (derived).
Cutting the bonus below the smallest bonus to save the rent
The rent cannot be reduced by paying less than b_star when no fee can be charged. With the weak indicator a bonus of 100 pounds raises the practice's expected bonus by 25 pounds for a review costing 30, so it skips the review and the purchaser pays 45 pounds per patient for indicators met by chance and buys no reviews (derived). Only a fee or an indicator that separates the two efforts better lowers the rent.
Sources
Laffont and Martimort on moral hazard with limited liability and a binary effort
Laffont JJ, Martimort D. The Theory of Incentives I. The Principal-Agent Model. Authors' manuscript dated 6 February 2001, deposited in Toulouse Capitole Publications, record 14941 (full text read). Published as The Theory of Incentives. The Principal-Agent Model. Princeton University Press; 2002. doi:10.1515/9781400829453 (not seen), so the section, equation and page numbers below are the manuscript's. Chapter 4, on moral hazard. Section 4.2.2 (Incentive Feasible Contracts), pages 151 and 152, writes the incentive constraint (4.3) as a weak inequality with the participation constraint (4.4) and Definition 4.1. Section 4.3 (Risk Neutrality and First-Best Implementation), pages 154 and 155, gives the transfers (4.12) and (4.13) with an expected payment equal to the disutility of effort psi, and Remark 2 notes that they make the incentive constraint exactly binding. Section 4.4 (The Trade-Off between Limited Liability Rent Extraction and Efficiency), pages 156 to 159, adds the limited liability constraints (4.17) and (4.18), and Proposition 4.2 gives, for a liability limit l at most pi_0 psi / Delta pi, the transfers (4.19) t SB = minus l and (4.20) t-bar SB = minus l plus psi / Delta pi and the expected limited liability rent (4.21) EU SB = minus l plus pi_0 psi / Delta pi, with no rent for a larger l. The text calls this rent the additional payment the principal incurs because of moral hazard combined with limited liability, and with l = 0 gives (4.22), the condition (4.23) for inducing effort, the second-best cost C SB = pi_1 psi / Delta pi, Figure 4.3 and Proposition 4.3. The records map c to psi, p_H to pi_1, p_L to pi_0 and ell to l.
Nobel committee background on the franchise contract and binding wealth constraints
Royal Swedish Academy of Sciences, Committee for the Prize in Economic Sciences in Memory of Alfred Nobel. Oliver Hart and Bengt Holmström: contract theory. Scientific background on the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2016. 10 October 2016 (full text read). Section 2.1 (Optimal Incentive Contracts), page 5, notes that payments between principal and agent are constrained by their financial resources, a constraint first set aside. On page 6, in the paragraph on hidden action and the classic moral-hazard model, a risk-neutral agent with sufficient financial resources is given a franchise contract, paying a fixed fee and receiving all of the realised benefit, so that it chooses the first-best action, and footnote 11 adds that managerial wealth constraints may be binding. The records follow this in letting a fee up to ell remove the rent (HE-EX-PAP-003).
Canonical Identity
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