Signature
q_eff = (b_0 - c) / b_1; r_star = 1 - alpha; P = (1 - r_star) * c * q_eff
| Inputs | Definition | Unit |
|---|---|---|
b_0 | Intercept of the linear marginal benefit of care | pounds per unit |
c | Constant cost of one more unit of care | pounds per unit |
b_1 | Slope of the linear marginal benefit of care | pounds per unit per unit |
alpha | Weight on a pound of patient benefit relative to a pound of net revenue, assumed known for the calculation | ratio |
q_eff | Units of care at which b(q) = c, so patient benefit net of cost is largest | bed-days per case |
|---|---|---|
r_star | Share of cost reimbursed that leads the provider to choose the efficient stay | proportion |
P | Fixed part of the blended payment that makes the payment for an efficient stay equal its cost | 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
Efficient stay, reimbursed share and fixed part from named cells
With BenIntercept, BenSlope, UnitCost and AgencyIndex named, the formulas return the efficient stay, the efficient reimbursed share and the budget-matched fixed part, held in EffDays, EffShare and BlendFixed.
=(BenIntercept-UnitCost)/BenSlope; =1-AgencyIndex; =(1-EffShare)*UnitCost*EffDays
Assumptions
Index of agency from 0 to 1 for the efficient share
alpha is positive and at most 1, so r_star runs from 0 (perfect agent) towards 1; an alpha above 1, a provider who overweights patient benefit, would call for a negative share, which a cost reimbursement rule cannot provide.
Zero margin on an efficient stay as the normalisation
Setting P so that payment equals cost at q_eff is the article's choice; any other fixed part leaves the chosen stay unchanged and only moves money between payer and provider.
Worked examples
Efficient blend for an index of agency of 0.5
The efficient stay is (1,200 minus 400) / 100, or eight days; half of cost is reimbursed, and the fixed part is 3,200 minus 0.5 times 400 times 8, or 1,600 pounds, as in the article.
b_0 = 1200; b_1 = 100; c = 400; alpha = 0.5; q_eff = 8; r_star = 0.5; P = 1600
Perfect agent needs pure case payment
With alpha equal to 1 no cost is reimbursed and the whole 3,200 pound cost of an eight-day stay is paid as a case price, as Ellis and McGuire state for a perfect agent.
b_0 = 1200; b_1 = 100; c = 400; alpha = 1; q_eff = 8; r_star = 0; P = 3200
Efficient blend for an index of agency of 0.8
One fifth of cost is reimbursed and the fixed part is 0.8 times 400 times 8, or 2,560 pounds (computed here for illustration).
b_0 = 1200; b_1 = 100; c = 400; alpha = 0.8; q_eff = 8; r_star = 0.2; P = 2560
Common errors
Treating the blend as better by construction
The blend restores the efficient stay only when r matches the provider's alpha; at alpha of 0.8 a half-reimbursed blend would give 9.5 days instead of eight (computed here for illustration), so a share chosen without evidence on agency can overshoot as well as undershoot.
Ignoring what the blend does to cost control
Partial reimbursement weakens the incentive to reduce the cost of each day, which the formula holds fixed; Ma and Mak report that cost reimbursement fails the internalisation principle for cost effort, so the efficient share for quantity is not efficient for cost.
Sources
Optimal supply-side cost sharing equals the index of agency 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 4: for efficiency b(q) = c is required, so the optimal r satisfies r = 1 minus a (eq. 14); the optimal cost sharing on the supply side (1 minus r) equals the index of agency; when a = 1 a perfect agent needs no additional hospital reimbursement; the prospective amount can be lowered as r is increased to maintain the same total payment per case.
Internalisation principle and cost effort in Ma and Mak
Ma CA, Mak HY. Incentives in healthcare payment systems. Oxford Research Encyclopedia of Economics and Finance. Oxford University Press; 2019. doi:10.1093/acrefore/9780190625979.013.61 (full text read). Introduction: a payment system achieves efficiency if and only if it gets the provider to internalise the social benefit and cost. Cost Reimbursement and Selection: when all variable cost is reimbursed the provider has no incentive to incur cost effort, so cost reimbursement fails the internalisation principle.
Canonical Identity
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