Signature
R = P + r * c * q; M = P - (1 - r) * c * q
| Inputs | Definition | Unit |
|---|---|---|
P | Fixed part of the payment, paid whatever care the case receives | pounds per case |
r | Proportion of the cost of care that the payer reimburses, from 0 (pure case payment) to 1 (full cost reimbursement) | proportion |
c | Cost of one more unit of care, such as a bed-day, taken as constant | pounds per unit |
q | Quantity of care, such as bed-days, given to one case | bed-days per case |
R | Payment received for one case that receives q units of care | pounds per case |
|---|---|---|
M | Revenue per case minus the cost of the care given, R minus c times q | pounds per case |
Function
Mixed provider payment and the quantity of care chosen under imperfect agency
Maps a provider payment rule with a prospective amount per case and a reimbursed share of cost to the provider's revenue and margin and, with the index of agency (the weight a provider puts on a pound of patient benefit relative to a pound of net revenue), to the quantity of care it chooses. Ellis and McGuire showed that the efficient reimbursed share equals one minus the index of agency. Notation follows the Contract Theory article and its blended bed-day example.
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Implementations
Excel
Revenue and margin per case from named payment parameters
With ProspAmt, ReimbShare, UnitCost and CareUnits named, the formulas return revenue per case and the provider margin, held in CaseRevenue and CaseMargin.
=ProspAmt+ReimbShare*UnitCost*CareUnits; =ProspAmt-(1-ReimbShare)*UnitCost*CareUnits
Assumptions
Linear cost of care in the mixed payment model
Total cost is c times q, the simple case Ellis and McGuire focus on; with rising marginal cost the reimbursed part becomes r times total cost and the margin no longer falls at a constant rate.
Payment based only on verifiable cost and case counts
Both parts of the payment depend on things the payer can verify, the case and its recorded cost; the patient benefit that the provider weighs (HE-FM-CTH-002) does not enter the payment.
Worked examples
Blend of 1,600 pounds plus half of cost for an eight-day stay
With a fixed part of 1,600 pounds, half of cost reimbursed and bed-days at 400 pounds, an eight-day stay brings 1,600 plus 1,600, or 3,200 pounds, exactly its cost, so the margin is zero, as in the article's table.
P = 1600; r = 0.5; c = 400; q = 8; R = 3200; M = 0
Pure case price of 3,200 pounds for a four-day stay
With no cost reimbursed the hospital receives 3,200 pounds whatever the stay; at four days the cost is 1,600 pounds and the margin 1,600 pounds, as in the article's table.
P = 3200; r = 0; c = 400; q = 4; R = 3200; M = 1600
Full cost reimbursement of a twelve-day stay
With no fixed part and all cost reimbursed, revenue equals the 4,800 pound cost of a twelve-day stay and the margin is zero, as in the article's table.
P = 0; r = 1; c = 400; q = 12; R = 4800; M = 0
Costly case of ten days under the blend
Two days beyond the eight-day stay cost 800 pounds; the blend reimburses half, so revenue rises to 3,600 pounds and the margin falls to minus 400 pounds. Under the pure case price two extra days would cut the margin by the full 800 pounds, as in the article.
P = 1600; r = 0.5; c = 400; q = 10; R = 3600; M = -400
Common errors
Cutting the case price to remove the margin
Lowering the pure case price from 3,200 to 1,600 pounds removes the margin on a four-day stay but leaves the hospital bearing every extra bed-day, so stays remain at four days; Ellis and McGuire note that recalibrating the prospective amount keeps the marginal incentive to reduce services and transfers the net gain to the payer and patients.
Reading r as the provider's share of cost
r is the share the payer reimburses; the provider's supply-side cost share is 1 minus r, which Ellis and McGuire set equal to the index of agency at the optimum (HE-FM-CTH-003). Swapping them is harmless only when r is 0.5; at r of 0.2 the provider bears 80 per cent of the cost of each extra day, not 20 per cent.
Sources
Mixed reimbursement revenue and net revenue in Ellis and McGuire
Ellis RP, McGuire TG. Journal of Health Economics. 1986;5(2):129-151. doi:10.1016/0167-6296(86)90002-0 (full text read). Section 2.2: revenue R(q) depends on the payment system, total cost is a constant c times quantity in the simple linear case. Section 4: the mixed system combines a prospective amount with a partial cost-based payment, R(q) = a + rC(q), the fraction of costs paid r; with linear cost, net revenue is a + (r minus 1)cq (eq. 12). Section 3.1: reducing the prospective payment to the new level maintains the marginal incentives to reduce supply but transfers the net gains to the payer and patients. Section 4: the optimal cost sharing on the supply side (1 minus r) equals the index of agency. (The paper's symbol for the prospective amount is written P here.)
Canonical Identity
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