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

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.

Signature

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
Inputs
InputsDefinitionUnit
aIntercept of the purchaser's linear marginal value S'(q) = a minus b q, the value of the first procedure, above theta_HGBP thousand per procedure
theta_LConstant marginal cost of the efficient supplierGBP thousand per procedure
bSlope of the linear marginal value, above zeroGBP thousand per procedure per procedure
theta_HConstant marginal cost of the inefficient supplier, above theta_LGBP thousand per procedure
nuPurchaser's prior probability of the low-cost type, at least 0 and below 1probability
Output
q_LOutput of the low-cost type, equal to its first-best output (a minus theta_L) / bprocedures per year
theta_vHigh cost plus nu / (1 minus nu) times the cost gap, the marginal cost the purchaser sets against marginal value for the high-cost typeGBP thousand per procedure
q_HOutput of the high-cost type at which marginal value equals the virtual cost, or zero when that is negativeprocedures per year
U_LRent left to the low-cost type, the cost gap times the high-cost type's outputGBP thousand per year
t_LCost of the low-cost type's output plus its information rentGBP thousand per year
t_HCost of the high-cost type's output, leaving it no rentGBP thousand per year

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.

Computational function

  • Computational function for the second-best menu against shut-down and complete information with an outside option

    Compares three policies for a purchaser buying from a supplier whose unit cost is theta_L with probability nu or theta_H otherwise, when both types have an outside option U_0 at least zero. Under complete information each type produces its first-best output and is paid its cost plus U_0. The second-best menu of HE-FM-MECH-002 pays the high-cost type its cost plus U_0 and the low-cost type its cost plus U_0 plus the rent (theta_H minus theta_L) q_H. Shutting out the high-cost type serves only the low-cost type at first-best output with no information rent. The purchaser takes the better of the two incentive-feasible policies, which is (2.32) of Laffont and Martimort, and the gap to the complete-information payoff measures what the hidden cost costs the purchaser (derived). The second-best output does not depend on U_0, so a large outside option makes shut-down optimal even when (2.28) has a positive solution.

    Inputs and outputs: a and b are the intercept and slope of the linear marginal value (GBP thousand per procedure, and per procedure per procedure), theta_L and theta_H the unit costs (GBP thousand per procedure), nu the probability of the low-cost type and U_0 the outside option of each type (GBP thousand per year). The chain returns the outputs q_L, q_FB and q_SB, the virtual cost theta_v, the values S_L, S_FB and S_SB of each output, the purchaser's expected payoffs V_FB (complete information), V_SB (second-best menu) and V_shut (shut-down), the chosen payoff V_star, the indicator shut (1 when shut-down is chosen) and loss, the expected cost of the hidden information (all payoffs GBP thousand per year).

    Assumption: As in HE-AS-MECH-005 to HE-AS-MECH-007, with S(q) = a q minus b q^2 / 2, a above theta_H and nu at least 0 and below 1, except that U_0 may be positive. Serving each type is assumed worthwhile under complete information (S_L minus theta_L q_L and S_FB minus theta_H q_FB both at least U_0), so V_FB serves both types. At a tie (2.32) holds with equality and the code reports shut-down, as the source's weak inequality does; the purchaser's payoff is the same either way.

    Worked example (equal priors, no outside option): With a = 10; b = 0.05; theta_L = 4; theta_H = 5; nu = 0.5; U_0 = 0 the outputs are q_L = 120; q_FB = 100; theta_v = 6; q_SB = 80, the values S_L = 840; S_FB = 750; S_SB = 640, and the payoffs V_FB = 305; V_SB = 260; V_shut = 180; V_star = 260; shut = 0; loss = 45. Of the loss of 45, 40 is the expected rent 0.5 x 1 x 80 and 5 is the expected efficiency loss 0.5 x (250 minus 240) on the high-cost type (derived).

    Worked example (outside option of 200): With U_0 = 200 and the other inputs as above, V_FB = 105; V_SB = 60; V_shut = 80; V_star = 80; shut = 1; loss = 25. The payoffs tie at U_0 = 160, where 240 less the rent term 80 equals U_0 in (2.32) (derived).

    Excel: With inputs in cells named ValueInterceptCF, ValueSlopeCF, CostLowCF, CostHighCF, ProbLowCF and OutsideOption, enter =(ValueInterceptCF-CostLowCF)/ValueSlopeCF in OutputLowCF, =(ValueInterceptCF-CostHighCF)/ValueSlopeCF in OutputFirstBestHigh, =MAX(0,(ValueInterceptCF-(CostHighCF+ProbLowCF/(1-ProbLowCF)*(CostHighCF-CostLowCF)))/ValueSlopeCF) in OutputHighCF, =ValueInterceptCF*OutputLowCF-ValueSlopeCF*OutputLowCF^2/2 in SurplusLow and the same form for OutputFirstBestHigh and OutputHighCF in SurplusFirstBestHigh and SurplusSecondBestHigh, then =ProbLowCF*(SurplusLow-CostLowCF*OutputLowCF-OutsideOption)+(1-ProbLowCF)*(SurplusFirstBestHigh-CostHighCF*OutputFirstBestHigh-OutsideOption) in PayoffFirstBest, =ProbLowCF*(SurplusLow-CostLowCF*OutputLowCF-OutsideOption-(CostHighCF-CostLowCF)*OutputHighCF)+(1-ProbLowCF)*(SurplusSecondBestHigh-CostHighCF*OutputHighCF-OutsideOption) in PayoffSecondBest, =ProbLowCF*(SurplusLow-CostLowCF*OutputLowCF-OutsideOption) in PayoffShutDown, =MAX(PayoffSecondBest,PayoffShutDown) in BestPayoff, =PayoffShutDown>=PayoffSecondBest in ShutDownChosen and =PayoffFirstBest-BestPayoff in CostOfHiddenCost.

    R: screen_menu <- function(a, b, thL, thH, nu, U0) { qL <- (a-thL)/b; qFB <- (a-thH)/b; tv <- thH+nu/(1-nu)*(thH-thL); qSB <- max(0, (a-tv)/b); S <- function(q) a*q-b*q^2/2; VFB <- nu*(S(qL)-thL*qL-U0)+(1-nu)*(S(qFB)-thH*qFB-U0); VSB <- nu*(S(qL)-thL*qL-U0-(thH-thL)*qSB)+(1-nu)*(S(qSB)-thH*qSB-U0); Vsh <- nu*(S(qL)-thL*qL-U0); c(qSB = qSB, VFB = VFB, VSB = VSB, Vshut = Vsh, Vstar = max(VSB, Vsh), shut = as.numeric(Vsh >= VSB), loss = VFB-max(VSB, Vsh)) } screen_menu(10, 0.05, 4, 5, 0.5, 0) returns qSB 80, VFB 305, VSB 260, Vshut 180, Vstar 260, shut 0 and loss 45, and sapply(seq(0, 250, 10), function(u) screen_menu(10, 0.05, 4, 5, 0.5, u)["shut"]) first returns 1 at U_0 = 160.

    Python: def screen_menu(a, b, thL, thH, nu, U0): S = lambda q: a*q-b*q**2/2; qL = (a-thL)/b; qFB = (a-thH)/b; tv = thH+nu/(1-nu)*(thH-thL); qSB = max(0.0, (a-tv)/b); VFB = nu*(S(qL)-thL*qL-U0)+(1-nu)*(S(qFB)-thH*qFB-U0); VSB = nu*(S(qL)-thL*qL-U0-(thH-thL)*qSB)+(1-nu)*(S(qSB)-thH*qSB-U0); Vsh = nu*(S(qL)-thL*qL-U0); Vst = max(VSB, Vsh); return dict(qSB=qSB, VFB=VFB, VSB=VSB, Vshut=Vsh, Vstar=Vst, shut=int(Vsh >= VSB), loss=VFB-Vst) screen_menu(10, 0.05, 4, 5, 0.5, 200) returns VFB 105, VSB 60, Vshut 80, Vstar 80, shut 1 and loss 25 (to floating point), and min(u for u in range(0, 260, 10) if screen_menu(10, 0.05, 4, 5, 0.5, u)['shut']) is 160.

    Test (equal priors, no outside option): PayoffSecondBest is compared with PayoffFirstBest less the expected rent ProbLowCF x (CostHighCF minus CostLowCF) x OutputHighCF and less the expected efficiency loss on the high-cost type, (1 minus ProbLowCF) times the drop in SurplusFirstBestHigh minus CostHighCF x OutputFirstBestHigh to SurplusSecondBestHigh minus CostHighCF x OutputHighCF, at a tolerance of 0.005. Expected result TRUE. FALSE shows a payoff that leaves out the rent (300), and doubling the PayoffSecondBest cell (520) or halving it (130) also gives FALSE. Excel check =ABS(PayoffSecondBest-(PayoffFirstBest-ProbLowCF*(CostHighCF-CostLowCF)*OutputHighCF-(1-ProbLowCF)*((SurplusFirstBestHigh-CostHighCF*OutputFirstBestHigh)-(SurplusSecondBestHigh-CostHighCF*OutputHighCF))))<0.005

    Test (outside option of 200): ShutDownChosen is TRUE and BestPayoff equals ProbLowCF times the low-cost type's surplus net of cost and outside option, 0.5 x (840 minus 480 minus 200) = 80, at a tolerance of 0.005. Expected result TRUE. FALSE shows the outside option left out of the payoffs, which keeps the high-cost type in (BestPayoff 260, ShutDownChosen FALSE), and doubling the BestPayoff cell (160) or halving it (40) also gives FALSE. Excel check =AND(ShutDownChosen,ABS(BestPayoff-ProbLowCF*(SurplusLow-CostLowCF*OutputLowCF-OutsideOption))<0.005)

    Common error (Outside option ignored): With a zero status quo and a positive solution of (2.28), serving the high-cost type is always better here, because S(q) less S'(q) q equals b q^2 / 2, which is above zero (derived, the argument of the remark in section 2.7.3 of Laffont and Martimort). Leaving U_0 out therefore always keeps the high-cost type in. With U_0 = 200 the menu of HE-EX-MECH-004 leaves the high-cost type 0 and the low-cost type 80 (GBP thousand), both below the outside option, so neither type accepts and nothing is bought (computed here).

    Source: Laffont JJ, Martimort D. The Theory of Incentives I. The Principal-Agent Model. Authors' manuscript dated 6 February 2001, Toulouse Capitole Publications, record 14941 (full text read); published by Princeton University Press, 2002, doi:10.1515/9781400829453, whose section numbers differ (see HE-SR-MECH-004). Manuscript chapter 2, section 2.7.3 (Shut-Down Policy), (2.30) to (2.32) and the remark on the status quo utility level U_0; section 2.7.1, Proposition 2.1.

    q_L = (a - theta_L) / b; q_FB = (a - theta_H) / b; theta_v = theta_H + nu / (1 - nu) * (theta_H - theta_L); q_SB = max(0, (a - theta_v) / b); S_L = a * q_L - b * q_L^2 / 2; S_FB = a * q_FB - b * q_FB^2 / 2; S_SB = a * q_SB - b * q_SB^2 / 2; V_FB = nu * (S_L - theta_L * q_L - U_0) + (1 - nu) * (S_FB - theta_H * q_FB - U_0); V_SB = nu * (S_L - theta_L * q_L - U_0 - (theta_H - theta_L) * q_SB) + (1 - nu) * (S_SB - theta_H * q_SB - U_0); V_shut = nu * (S_L - theta_L * q_L - U_0); V_star = max(V_SB, V_shut); shut = (V_shut >= V_SB); loss = V_FB - V_star

Try this function

Implementations

  • Excel

    Excel second-best two-type menu from a linear marginal value

    With the intercept and slope of the marginal value in cells named ValueIntercept and ValueSlope, the unit costs in CostLowSB and CostHighSB and the probability of the low-cost type in ProbLowSB, the formulas return OutputLowSB, VirtualCostSB, OutputHighSB, RentLowSB, TransferLowSB and TransferHighSB in that order.

    =(ValueIntercept-CostLowSB)/ValueSlope; =CostHighSB+ProbLowSB/(1-ProbLowSB)*(CostHighSB-CostLowSB); =MAX(0,(ValueIntercept-VirtualCostSB)/ValueSlope); =(CostHighSB-CostLowSB)*OutputHighSB; =CostLowSB*OutputLowSB+RentLowSB; =CostHighSB*OutputHighSB

Assumptions

  • Linear marginal value of output in the second-best menu

    The purchaser's value is S(q) = a q minus b q^2 / 2, so the marginal value a minus b q falls in a straight line, with a above theta_H so that both first-best outputs are positive. This functional form is chosen here for a closed form; Laffont and Martimort require only S' above zero, S'' below zero and S(0) = 0, and (2.26) and (2.28) hold for any such S.

  • Risk-neutral purchaser offering the menu after the supplier learns its cost

    The purchaser maximises expected S(q) less the transfer with the prior nu at least 0 and below 1, offers the menu once the supplier knows its type (the timing of Figure 2.1 of Laffont and Martimort), and both sides are committed to the contract. The supplier is risk neutral.

  • Zero outside option and no fixed cost in the second-best menu

    Both types have a status quo utility of zero and no fixed cost, the setting of Proposition 2.1. With these, and a positive solution of (2.28), serving the high-cost type is better than shutting it out (the remark in section 2.7.3 of Laffont and Martimort, which holds for any value function with S'' below zero; checked here for the linear form on a grid of 777 cases); a positive outside option is handled in HE-CF-MECH-001.

Worked examples

  • Second-best menu for elective procedures with equal priors

    Illustrative figures. The first procedure is worth GBP 10,000 to the commissioner and each further one GBP 50 less; the provider's unit cost is GBP 4,000 or GBP 5,000 with equal probability. The low-cost type produces its first-best 120 procedures. The virtual cost of the high-cost type is 5 + 1 x 1 = 6, so it produces 80 rather than its first-best 100, and the low-cost type's rent is 1 x 80 = 80 (GBP 80,000). The transfers are GBP 560,000 and GBP 400,000.

    a = 10; b = 0.05; theta_L = 4; theta_H = 5; nu = 0.5; q_L = 120; theta_v = 6; q_H = 80; U_L = 80; t_L = 560; t_H = 400

  • Second-best menu when the low-cost type is believed absent

    Limiting case with the same illustrative figures and nu = 0. The virtual cost equals the high cost, so the high-cost type produces its first-best 100 procedures and the formula leaves the low-cost type a rent of 100 (GBP 100,000) on an output it is believed never to supply.

    a = 10; b = 0.05; theta_L = 4; theta_H = 5; nu = 0; q_L = 120; theta_v = 5; q_H = 100; U_L = 100; t_L = 580; t_H = 500

  • Second-best menu that shuts out the high-cost type

    Illustrative figures as above with nu = 0.9. The virtual cost is 5 + 9 x 1 = 14, above the value of the first procedure, so (2.28) has no positive solution, the high-cost type is offered nothing and the low-cost type is paid its cost of GBP 480,000 with no rent, the shut-down case of Laffont and Martimort.

    a = 10; b = 0.05; theta_L = 4; theta_H = 5; nu = 0.9; q_L = 120; theta_v = 14; q_H = 0; U_L = 0; t_L = 480; t_H = 0

Common errors

  • Prior probability used in place of the odds in the virtual cost

    The distortion term is nu / (1 minus nu) times the cost gap, the odds of the low-cost type. Writing nu alone gives a virtual cost of 5.5 in place of 6 with equal priors, a high-cost output of 90 in place of 80 and a rent of GBP 90,000 in place of GBP 80,000 (computed here); with nu = 0.9 it gives 5.9 and an output of 82 where the correct menu shuts the high-cost type out.

  • Distorting the low-cost type's output as well

    Laffont and Martimort's solution leaves the low-cost type's output at first best, since its rent depends only on q_H (2.26). Setting q_L where marginal value equals theta_L plus the same distortion term, 4 + 1 = 5, gives 100 procedures in place of 120 and lowers the commissioner's expected payoff from 260 to 255 (GBP thousand) with no saving in rent (computed here).

  • Low-cost transfer that leaves out the information rent

    Paying the low-cost type only its cost, t_L = 4 x 120 = 480, with the high-cost contract of 80 procedures for GBP 400,000 makes the low-cost type better off on the high-cost contract by GBP 80,000, so it mimics and the menu fails HE-FM-MECH-001 (IC_L of minus 80, computed here).

Sources

  • Laffont and Martimort on the second-best menu and its information rent

    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 2.6 (The Optimization Program of the Principal), the principal's problem of maximising nu (S(q) minus t) + (1 minus nu)(S(q-bar) minus t-bar) subject to (2.9) to (2.12), rewritten as expected allocative efficiency less expected information rent in (2.19). Section 2.7.1 (The Asymmetric Information Optimal Contract), (2.24) and (2.25), the efficient type's incentive constraint and the inefficient type's participation constraint bind; (2.26), no distortion for the efficient type; (2.27); and Proposition 2.1, with (2.28) S'(q-bar SB) = theta-bar + nu / (1 minus nu) Delta theta, (2.29) the rent Delta theta q-bar SB, and the transfers theta q* + Delta theta q-bar SB and theta-bar q-bar SB. The closed form for a linear marginal value is derived here.

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  • Laffont and Martimort on first-best outputs and the shut-down policy

    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 2.3.1 (First-Best Production Levels), (2.4) and (2.5), first-best outputs where marginal value equals each type's marginal cost. Section 2.7.3 (Shut-Down Policy), the high-cost output is set to zero when (2.28) has no positive solution; (2.31), shut-down is optimal when nu Delta theta q-bar SB is at least (1 minus nu) times S(q-bar SB) minus theta-bar q-bar SB; and (2.32), the same condition with a status quo utility U_0 above zero for both types.

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Canonical Identity

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