Functions & Formulae

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

Designer's problem of a direct mechanism with incentive and participation constraints in its two-type screening form

m(S, theta_L, theta_H, nu) = (q_L, q_H, t_L, t_H, U_L)

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.

  • Incentive and participation slacks of a two-type procurement menu

    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)

    A purchaser offers two contracts, (t_L, q_L) intended for a low-cost supplier and (t_H, q_H) for a high-cost one, and cannot observe which type it faces. Each type's information rent is its transfer less its cost on its own contract, and each incentive slack is that rent less what the type would earn by taking the other contract at its own cost. The menu is incentive feasible when both incentive slacks and both rents are at least zero, inequalities (2.9) to (2.12) of Laffont and Martimort. These are the article's incentive compatibility and participation constraints for one participant with two types and an outside option of zero. A low-cost type that takes the high-cost contract earns the high-cost type's rent plus (theta_H minus theta_L) q_H, which is why a menu that leaves both types no rent fails. Each rent takes the form of the margin in HE-FM-CTH-001 with no reimbursed share (derived), there computed per case of hospital care; the incentive slacks, which value one type's cost on the other type's contract, have no live counterpart.

  • Second-best two-type procurement menu with a linear marginal value of output

    q_L = (a - theta_L) / b; theta_v = theta_H + nu / (1 - nu) * (theta_H - theta_L); q_H = max(0, (a - theta_v) / b); U_L = (theta_H - theta_L) * q_H; t_L = theta_L * q_L + U_L; t_H = theta_H * q_H

    The purchaser values output at S(q) and, knowing only that the supplier is low-cost with probability nu, chooses the menu that maximises expected S(q) less the transfer subject to the constraints of HE-FM-MECH-001. In Laffont and Martimort's solution the low-cost type produces its first-best output, the high-cost type's output is set where the marginal value equals the virtual cost theta_H + nu / (1 minus nu) (theta_H minus theta_L), and the low-cost type receives the information rent (theta_H minus theta_L) q_H while the high-cost type receives none. With a linear marginal value S'(q) = a minus b q, an illustrative form chosen here, both outputs solve in closed form (derived). The output step takes the form of HE-FM-CTH-002 and the first-best output that of HE-FM-CTH-003, which compute stays of hospital care; here the cost fed in is the virtual cost of (2.28). The max function sets q_H to zero when (2.28) has no positive solution, the authors' shut-down case.

Mechanism Design — Functions & Formulae | HealthEconomics.wiki