Signature
p_bar = k * Delta_E - Delta_C_o
| Inputs | Definition | Unit |
|---|---|---|
k | Cost-effectiveness threshold the payer applies to funding decisions | currency per QALY |
Delta_E | Incremental QALYs per patient from the product compared with current care | QALYs per patient |
Delta_C_o | Incremental cost per patient other than the product's price, such as administration costs | currency per patient |
p_bar | Highest price per patient at which the product's incremental cost-effectiveness ratio does not exceed the threshold | currency per patient |
|---|
Function
Nash bargaining surplus-division function for negotiated health care prices
Maps the payoffs two negotiating parties obtain from each feasible agreement, their disagreement payoffs and their relative bargaining power to the agreement that maximises the weighted product of their gains over disagreement. In a price negotiation between a payer and a manufacturer the feasible agreements are prices from the manufacturer's reservation price up to the payer's maximum price, and the function returns the price that splits the surplus between them in proportion to their bargaining weights. A weight of one half gives Nash's original symmetric solution. The payer's maximum price can come from a cost-effectiveness threshold (HE-FM-BARG-002), the manufacturer's weight from the parties' patience in an alternating-offers model (HE-FM-BARG-003), and an outside option changes the result only when it binds (HE-FM-BARG-004).
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Implementations
Excel
Payer's maximum price from a threshold in one cell
Excel multiplies the threshold by the incremental QALYs and subtracts the other incremental costs, using named cells per patient.
=Threshold*IncQALYs-OtherIncCost
Assumptions
One course per patient priced as a unit
Each patient receives one course, so the price, the QALY gain and the other costs are all per patient. If a course is supplied in several packs, the maximum price per pack is p_bar divided by the number of packs per patient.
Threshold that reflects the health displaced by spending
The payer funds the product when the ratio is at or below k, Delta_E is above zero, and k represents the health displaced by extra spending. If k is above the value of the health displaced, a price at p_bar displaces more health than the product adds.
Worked examples
Maximum price at £20,000 per QALY for an illustrative treatment
A gain of 0.5 QALYs per patient, other incremental costs of £2,000 and a threshold of £20,000 per QALY give a maximum price of £8,000 per patient, as in the article. With a reservation price of £1,000 the surplus to divide is £7,000 per patient.
k = 20000; Delta_E = 0.5; Delta_C_o = 2000; p_bar = 8000
Maximum price at an illustrative £30,000 per QALY
For the same treatment a threshold of £30,000 per QALY raises the maximum price to £13,000 per patient, so a higher threshold widens the range over which the price is bargained.
k = 30000; Delta_E = 0.5; Delta_C_o = 2000; p_bar = 13000
Common errors
Leaving other incremental costs out of the maximum price
Multiplying the threshold by the QALY gain alone gives £10,000 rather than £8,000 per patient. At £10,000 the ratio is £24,000 per QALY, above the £20,000 threshold, so the payer would not fund the product at that price.
Applying the maximum price per patient to each pack
When a course is supplied in several packs, setting each pack at p_bar multiplies the cost per patient. With four packs per course and a maximum of £8,000 per patient, the maximum price per pack is £2,000.
Sources
Claxton and colleagues on the price at which the ratio equals the threshold
Claxton K, Briggs A, Buxton MJ, Culyer AJ, McCabe C, Walker S, Sculpher MJ. Value based pricing for NHS drugs: an opportunity not to be missed? BMJ. 2008;336(7638):251-254. Box 1, in which a price at which the ratio equals the threshold leaves net health benefit to the NHS at zero and a higher price is not cost effective, and the text noting that such a price passes all the benefit of the innovation to the manufacturer as revenue.
Berdud and colleagues on the threshold as the payer's maximum ratio
Berdud M, Ferraro J, Towse A. A theory on ICER pricing and optimal levels of cost-effectiveness thresholds: a bargaining approach. Frontiers in Health Services. 2023;3:1055471. Section 2.3, in which the payer commits to a threshold before bargaining and, when the developer holds all the bargaining power, the agreed ratio equals the health system's maximum ability to pay.
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