Signature
U_L = t_L - theta_L * q_L; U_H = t_H - theta_H * q_H; IC_L = U_L - (t_H - theta_L * q_H); IC_H = U_H - (t_L - theta_H * q_L); feasible = (IC_L >= 0) and (IC_H >= 0) and (U_L >= 0) and (U_H >= 0)
| Inputs | Definition | Unit |
|---|---|---|
t_L | Payment in the contract intended for the low-cost supplier | GBP thousand per year |
theta_L | Constant marginal cost of the efficient supplier, known to the supplier only | GBP thousand per procedure |
q_L | Quantity in the contract intended for the low-cost supplier, such as elective procedures | procedures per year |
t_H | Payment in the contract intended for the high-cost supplier | GBP thousand per year |
theta_H | Constant marginal cost of the inefficient supplier, above theta_L | GBP thousand per procedure |
q_H | Quantity in the contract intended for the high-cost supplier | procedures per year |
U_L | Transfer less cost for the low-cost type taking its own contract, t_L minus theta_L q_L | GBP thousand per year |
|---|---|---|
U_H | Transfer less cost for the high-cost type taking its own contract, t_H minus theta_H q_H | GBP thousand per year |
IC_L | Rent of the low-cost type on its own contract less what it earns on the high-cost contract; negative when it prefers to mimic | GBP thousand per year |
IC_H | Rent of the high-cost type on its own contract less what it earns on the low-cost contract; negative when it prefers to mimic | GBP thousand per year |
feasible | 1 when both incentive slacks and both rents are at least zero, otherwise 0 | indicator (1 or 0) |
Function
Designer's problem of a direct mechanism with incentive and participation constraints in its two-type screening form
Maps the designer's objective, the distribution of participants' private types and a candidate allocation rule x(theta) and payment rule t(theta) to the designer's expected objective, subject to incentive compatibility, under which each type does at least as well reporting truthfully as mimicking any other type, and participation, under which each type does at least as well as by staying out, as in the article's display. The general problem is an optimisation over functions and has no calculator form. Its simplest instance, one supplier whose unit cost is low (theta_L) or high (theta_H) and a purchaser who offers a menu of output and transfer pairs, is the screening model of Laffont and Martimort, in which the allocation is a quantity q and the payment a transfer t. HE-FM-MECH-001 checks whether a given menu satisfies the four constraints and reports the information rents, HE-FM-MECH-002 gives the purchaser's second-best menu with a linear marginal value of output, and HE-CF-MECH-001 compares that menu with shutting out the high-cost type and with the complete-information benchmark when the supplier has a positive outside option. A type's payoff on its own contract, transfer less cost times quantity, takes the form of the margin in HE-FM-CTH-001 with no reimbursed share, which computes it per case of hospital care. The Ellis and McGuire records HE-FN-CTH-001 to HE-CF-CTH-001 set a payment rule for a provider whose parameters are known, with no hidden type. The article's tender profit of GBP 60,000, a margin of GBP 0.60 per pack times 100,000 packs, is a price margin times a volume, the form of HE-FM-CSKM-002 with one risk group, and the second-price rule that sets the price belongs to the auction-theory-health concept, so neither is restated here. Matching and kidney exchange rules are algorithms on a compatibility graph and have no formula record. Notation follows the Mechanism Design article for the general problem and Laffont and Martimort for the two-type case.
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Implementations
Excel
Excel screening menu check with rents and incentive slacks
With the menu in cells named TransferLow, OutputLow, TransferHigh and OutputHigh and the two unit costs in CostLow and CostHigh, the formulas return the rents in cells named RentLow and RentHigh, the incentive slacks in IcSlackLow and IcSlackHigh, and TRUE in MenuFeasible when all four are at least zero. The ROUND to six decimals stops a binding constraint failing on floating-point residue when costs such as 4.6 are typed.
=TransferLow-CostLow*OutputLow; =TransferHigh-CostHigh*OutputHigh; =RentLow-(TransferHigh-CostLow*OutputHigh); =RentHigh-(TransferLow-CostHigh*OutputLow); =AND(ROUND(IcSlackLow,6)>=0,ROUND(IcSlackHigh,6)>=0,ROUND(RentLow,6)>=0,ROUND(RentHigh,6)>=0)
Assumptions
Two cost types with constant marginal cost in the screening check
The supplier's cost is theta q with theta equal to theta_L or theta_H, as in (2.1) and (2.2) of Laffont and Martimort, with the fixed cost set to zero as the authors do after (2.6). A supplier with more than two types, or with a cost that is not linear in output, needs the constraints between every pair of types.
Outside option of zero for both supplier types
The participation constraints (2.11) and (2.12) compare each rent with zero, the authors' normalisation of the status quo utility level. A supplier with a positive outside option U_0 needs each rent to be at least U_0, which HE-CF-MECH-001 handles.
Weak inequalities and indifferent types in the screening check
Definition 2.1 of Laffont and Martimort requires each type to weakly prefer its own contract, so a slack of exactly zero passes and an indifferent type is taken to choose the contract intended for it. The authors note that strict preference can be obtained at an arbitrarily small cost to the purchaser.
Observable and enforceable quantity and transfer
Laffont and Martimort take the quantity and the transfer to be observable and verifiable by a third party, so each contract can be enforced as written. Only the supplier's cost is hidden.
Worked examples
Second-best procurement menu passes every screening constraint
Illustrative figures. A commissioner buys elective procedures from a provider whose unit cost is GBP 4,000 or GBP 5,000 and offers 120 procedures for GBP 560,000 or 80 procedures for GBP 400,000, the second-best menu of HE-EX-MECH-004. The low-cost type earns a rent of 80 (GBP 80,000) and would earn the same 80 on the other contract, so its incentive constraint binds; the high-cost type earns no rent and would lose 40 by taking the larger contract. All four constraints hold.
t_L = 560; q_L = 120; t_H = 400; q_H = 80; theta_L = 4; theta_H = 5; U_L = 80; U_H = 0; IC_L = 0; IC_H = 40; feasible = 1
First-best procurement menu fails the low-cost type's incentive constraint
Illustrative figures. Paying each type exactly its cost for its efficient output, 120 procedures for GBP 480,000 or 100 for GBP 500,000, leaves both types no rent and satisfies both participation constraints. The low-cost type would earn 100 (GBP 100,000) by taking the high-cost contract, which is (5 minus 4) x 100, so IC_L is minus 100 and the menu is not incentive feasible, the failure Laffont and Martimort show for their first-best contracts.
t_L = 480; q_L = 120; t_H = 500; q_H = 100; theta_L = 4; theta_H = 5; U_L = 0; U_H = 0; IC_L = -100; IC_H = 120; feasible = 0
Pooling contract passes the screening constraints with a rent to the low-cost type
Illustrative figures. Offering both types the same contract, 100 procedures for GBP 500,000, satisfies both incentive constraints trivially, a pooling contract in the authors' terms. The high-cost type's participation constraint is the one that binds, and the low-cost type keeps a rent of 100 (GBP 100,000).
t_L = 500; q_L = 100; t_H = 500; q_H = 100; theta_L = 4; theta_H = 5; U_L = 100; U_H = 0; IC_L = 0; IC_H = 0; feasible = 1
Common errors
Checking participation only in a screening menu
A menu that pays each type its cost, such as 120 procedures for GBP 480,000 and 100 for GBP 500,000, passes both participation constraints, yet the low-cost type takes the second contract and earns GBP 100,000 (computed here). The purchaser then buys 100 procedures for GBP 500,000 from a supplier whose cost is GBP 400,000, so a menu is checked against all four constraints.
Mimicking payoff evaluated at the wrong type's cost
The incentive constraint of the low-cost type values the high-cost contract at the low cost, t_H minus theta_L q_H. Using theta_H there turns IC_L into U_L minus U_H, which is 0 for the first-best menu of HE-EX-MECH-002, so a menu that fails by 100 (GBP 100,000) is passed as feasible (computed here).
Strict inequality used for a binding screening constraint
The purchaser's optimal menu leaves the low-cost type exactly indifferent, IC_L = 0 in HE-EX-MECH-001. Testing IC_L > 0 in place of IC_L >= 0 rejects the second-best menu, and with costs typed as 4.0 and 4.6 a binding slack can carry a floating-point residue (about 6E-14 in a Python check of the second-best menu for those costs), which the ROUND in HE-IM-MECH-001 removes.
Sources
Laffont and Martimort on incentive and participation constraints for two types
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). The printed book numbers the chapter 2 sections one lower than the manuscript (its section 2.3 is Incentive Feasible Menu of Contracts) and its equation numbers were not checked, so the section and equation numbers below are the manuscript's. Chapter 2, section 2.4.1 (Incentive Compatibility and Participation), Definition 2.1 and inequalities (2.9) and (2.10), the incentive constraints t minus theta q at least t-bar minus theta q-bar for the efficient type and the reverse for the inefficient type; inequalities (2.11) and (2.12), the participation constraints with a zero status quo; Definition 2.2, a menu is incentive feasible when (2.9) to (2.12) hold; footnote 8 on weak preference. Section 2.5 (Information Rents), (2.18), the efficient type's payoff from mimicking equals the inefficient type's rent plus Delta theta times the inefficient type's output. Sections 2.2.1 and 2.2.2 set the linear cost, the two types and the verifiable quantity and transfer.
Nobel committee background on incentive compatibility and the participation constraint
Royal Swedish Academy of Sciences. Mechanism design theory. Scientific background on the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2007. 2007 (full text read). Section 2.1 (Incentive compatibility and the revelation principle) states Hurwicz's definition of an incentive-compatible direct mechanism, adds a participation constraint that no agent be made worse off by participating, and states that by the revelation principle much of the literature seeks a direct mechanism that maximises the goal function subject to the incentive-compatibility constraint and, where appropriate, the participation constraint, the problem the article displays.
Canonical Identity
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