Economically justifiable price per unit and per patient at one threshold

Signature

EJP_patient = lambda * Delta_E - Delta_C_other; EJP = EJP_patient / U

Function

Economically justifiable price function solving the ICER for the unit price

Computational function

• Computational function: probabilistic economically justifiable price at a stated probability of cost-effectiveness

  Takes matched draws from a probabilistic sensitivity analysis, a threshold and a required probability of cost-effectiveness, and returns two prices per unit. EJP_mean is the price at which expected incremental net benefit is zero, the probabilistic EJP of the article. EJP_alpha is the highest price at which at least a share alpha of the draws is cost-effective at lambda, the stricter reading the article describes. Because each draw's net benefit is linear in price, draw i is cost-effective at a price P exactly when P is at or below its own EJP_i, so the probability of cost-effectiveness at P is the share of draw-level EJPs at or above P and the search over prices needs no further model runs. The inputs differ from the formula's variables: the function takes vectors of draws and a probability level and returns an order statistic as well as a closed-form value.

  Inputs and outputs: Delta_E_i: Incremental QALYs per patient in draw i of N matched draws; required. Unit: QALYs per patient.; Delta_C_other_i: Incremental cost per patient in draw i excluding the medicine's acquisition cost; required. Unit: money per patient.; U_i: Discounted units per patient in draw i, above zero, or one value when units do not vary; required. Unit: units per patient.; lambda: Threshold; required, above zero. Unit: money per QALY.; alpha: Required probability of cost-effectiveness, for example 0.5 or 0.8; required, above zero and at most 1. Unit: probability.; EJP_i: EJP of draw i from HE-FM-EJP-001. Unit: money per unit.; EJP_mean: Price per unit at which mean incremental net benefit is zero. Unit: money per unit.; EJP_alpha: Highest price per unit at which the share of cost-effective draws is at least alpha, the k-th largest EJP_i with k equal to alpha times N rounded up. Unit: money per unit.

  Assumption: Price enters each draw's incremental cost linearly as P times U_i, and the draws come from one probabilistic run with the price held out. A draw counts as cost-effective when its incremental net benefit is zero or above (for a draw with a positive QALY gain, an ICER at or below lambda); with continuous draws this matches the strict rule of HE-FM-CEAC-001 except at ties. Where price changes other model quantities, the probabilistic analysis is repeated at several prices, as the article describes, and the share of cost-effective draws is read at each.

  Worked example (Five illustrative draws at the 80% level): Five draws (computed here for illustration) with QALY gains of 0.56, 0.66, 0.76, 0.86 and 0.96, other costs of GBP 2,100, 1,900, 1,700, 1,500 and 1,300 and 25 packs have the article's means of 0.76 and GBP 1,700. At GBP 25,000 per QALY their draw-level EJPs are GBP 476, 584, 692, 800 and 908 per pack. Expected net benefit is zero at GBP 692, the article's probabilistic EJP, but Drug X is cost-effective in at least 80% of draws only up to GBP 584. lambda = 25000; alpha = 0.8; N = 5; k = 4; EJP_mean = 692; EJP_alpha = 584

  Worked example (Same draws at the 50% level): At a required probability of 0.5, k is 3 and EJP_alpha is the third largest draw-level EJP, GBP 692; with these symmetric draws the median and the mean coincide. lambda = 25000; alpha = 0.5; N = 5; k = 3; EJP_mean = 692; EJP_alpha = 692

  Excel: =(Threshold*AVERAGE(DeltaE)-AVERAGE(DeltaCOther))/AVERAGE(Units) returns EJP_mean and =LARGE((Threshold*DeltaE-DeltaCOther)/Units,ROUNDUP(Alpha*ROWS(DeltaE)-1E-9,0)) returns EJP_alpha, with DeltaE, DeltaCOther and Units as matched columns of draws (Units may be one cell) and Threshold and Alpha as single cells. Excel 365 evaluates the array inside LARGE directly; earlier versions need it entered as an array formula. The small offset stops a product such as 0.7 times 10 being rounded up to 8.

  R: ejp_psa <- function(dE, dC_other, U, lambda, alpha) { ejp_i <- (lambda*dE-dC_other)/U; k <- ceiling(alpha*length(dE)-1e-9); list(EJP_mean = (lambda*mean(dE)-mean(dC_other))/mean(U), EJP_alpha = sort(ejp_i, decreasing = TRUE)[k]) } Base R; ejp_psa(c(0.56, 0.66, 0.76, 0.86, 0.96), c(2100, 1900, 1700, 1500, 1300), 25, 25000, 0.8) returns 692 and 584.

  Python: def ejp_psa(dE, dC_other, U, lam, alpha): dE, dC_other, U = (np.asarray(x, dtype=float) for x in (dE, dC_other, U)); ejp_i = (lam*dE-dC_other)/U; k = math.ceil(alpha*dE.size-1e-9); return {"EJP_mean": float((lam*dE.mean()-dC_other.mean())/U.mean()), "EJP_alpha": float(np.sort(ejp_i)[::-1][k-1])} Needs import math and import numpy as np; lam is the threshold, because lambda is reserved in Python. Returns the same values as the R function.

  Test (Share of cost-effective draws at EJP_alpha): In the five-draw example 4 of 5 draws are cost-effective at GBP 584 per pack and 3 of 5 at GBP 585. Expected result: TRUE. Excel check: =AND(SUMPRODUCT(--({476,584,692,800,908}>=584))/5>=0.8,SUMPRODUCT(--({476,584,692,800,908}>=585))/5<0.8)

  Test (EJP_mean reproduces the article's probabilistic EJP): The means of 0.76 QALYs and GBP 1,700 with 25 packs return GBP 692 per pack. Expected result: TRUE. Excel check: =ROUND((25000*0.76-1700)/25,2)=692

  Common error (Averaging draw-level EJPs when units vary): Two illustrative draws with 0.80 QALYs and GBP 1,500 of other costs but 20 and 30 packs have draw-level EJPs of GBP 925 and about GBP 616.67, whose mean is about GBP 770.83. Expected net benefit is zero at GBP 740, the GBP 18,500 per patient divided by the mean of 25 packs (computed here for illustration).

  Common error (Assuming the deterministic EJP is cost-effective with high probability): Zeevat and colleagues found that at their base-case EJP of EUR 16.38 for a hypothetical RSV vaccine at EUR 20,000 per QALY, the probability of cost-effectiveness in the Netherlands was 0.44, and at GBP 72.29 at GBP 20,000 per QALY in the United Kingdom it was 0.59. A price chosen to reach a higher probability lies below the EJP.

  Source: National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; published 31 January 2022, last updated 31 March 2026. Section 4.7.12 (preferred estimate from probabilistic analysis unless the model is linear) and section 4.7.15 (present the probability that the treatment is cost effective at maximum acceptable ICERs of GBP 25,000 to GBP 35,000 per QALY gained); Zeevat F, Luttjeboer J, Paulissen JHJ, van der Schans J, Beutels P, Boersma C, Postma MJ. Exploratory analysis of the economically justifiable price of a hypothetical RSV vaccine for older adults in the Netherlands and the United Kingdom. Journal of Infectious Diseases. 2022;226(Suppl 1):S102-S109. doi:10.1093/infdis/jiab118. Results of the probabilistic sensitivity analysis.

  EJP_i = (lambda * Delta_E_i - Delta_C_other_i) / U_i; EJP_mean = (lambda * sum_(i=1)^N [Delta_E_i] - sum_(i=1)^N [Delta_C_other_i]) / sum_(i=1)^N [U_i]; P_CE(P) = (1/N) * sum_(i=1)^N [1(EJP_i >= P)]; EJP_alpha = max {P : P_CE(P) >= alpha}

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Implementations

• Excel

  EJP per unit from named threshold, QALY, cost and unit cells

  With the threshold in a cell named Threshold, the incremental QALYs in DeltaE, the other incremental cost in DeltaCOther and the discounted units per patient in Units, Excel returns the EJP per unit.

  =(Threshold*DeltaE-DeltaCOther)/Units

• Excel

  Per-patient EJP from named threshold, QALY and cost cells

  The same named cells without the division by Units return the per-patient EJP.

  =Threshold*DeltaE-DeltaCOther

Assumptions

• Price enters the incremental cost linearly in an EJP calculation

  The acquisition cost per patient is the unit price times U, and neither Delta_E nor Delta_C_other changes with the price. Where price affects other model quantities, the price is iterated in the model until the ICER equals the threshold, as the article describes, instead of using the closed form.

• Inputs of an EJP from one comparator, perspective and discount rate

  Delta_E, Delta_C_other and U come from the same comparator, population, perspective and time horizon, with costs and health effects discounted at the same rate, 3.5% a year in NICE's reference case (PMG36 section 4.5.1), so later units count for less. Delta_E is above zero; the formula is not read for a medicine that loses QALYs.

• EJP compared with the net price paid

  The EJP is compared with the price the payer would actually pay, after patient access schemes and other commercial arrangements, which NICE's reference case uses (PMG36 section 4.4.4), not with the list price.

Worked examples

• EJP for Drug X per pack at 25,000 pounds per QALY

  In the article's illustrative Drug X the model gives 0.80 incremental QALYs, GBP 1,500 of other incremental costs (GBP 4,000 of extra monitoring and administration less GBP 2,500 of avoided hospital care) and 25 discounted packs per patient. At GBP 25,000 per QALY the per-patient EJP is GBP 18,500 and the EJP is GBP 740 per pack.

  lambda = 25000; Delta_E = 0.80; Delta_C_other = 1500; U = 25; EJP_patient = 18500; EJP = 740

• EJP for Drug X per pack at 35,000 pounds per QALY

  At GBP 35,000 per QALY the same inputs give a per-patient EJP of GBP 26,500 and an EJP of GBP 1,060 per pack, as in the article; only the threshold has changed.

  lambda = 35000; Delta_E = 0.80; Delta_C_other = 1500; U = 25; EJP_patient = 26500; EJP = 1060

• EJP for Drug X when treatment lasts 30 packs

  If patients stayed on Drug X for 30 discounted packs with the same QALY gain, the per-patient EJP would stay at GBP 18,500 but the EJP per pack would fall to about GBP 616.67, because the same value is spread over more units.

  lambda = 25000; Delta_E = 0.80; Delta_C_other = 1500; U = 30; EJP_patient = 18500; EJP = 616.67

• Probabilistic EJP for Drug X from mean PSA outputs

  If probabilistic analysis gave a mean QALY gain of 0.76 and a mean other cost of GBP 1,700, with 25 packs, the price at which expected incremental net benefit is zero would be GBP 692 per pack, as in the article. Because net benefit is linear in price, the formula applied to the means of the probabilistic outputs gives this price; in a non-linear model it differs from the EJP computed at mean inputs.

  lambda = 25000; Delta_E = 0.76; Delta_C_other = 1700; U = 25; EJP_patient = 17300; EJP = 692

• Per-patient EJP for Drug X with one unit per patient

  Entering 1 for U returns the per-patient EJP as the price: GBP 18,500 at GBP 25,000 per QALY, the most the payer could spend on Drug X for each patient.

  lambda = 25000; Delta_E = 0.80; Delta_C_other = 1500; U = 1; EJP_patient = 18500; EJP = 18500

• Negative EJP when other costs exceed the value of the QALY gain

  With a QALY gain of only 0.05 and the same GBP 1,500 of other incremental costs, the value of the gain at GBP 25,000 per QALY is GBP 1,250, so the per-patient EJP is minus GBP 250 and the EJP minus GBP 10 per pack (computed here for illustration). Mladsi and colleagues call this the zero-price conundrum: costs the medicine cannot offset, not a modelling error.

  lambda = 25000; Delta_E = 0.05; Delta_C_other = 1500; U = 25; EJP_patient = -250; EJP = -10

Common errors

• Adding the other incremental costs in the EJP

  Adding Delta_C_other to lambda times Delta_E instead of subtracting it gives GBP 860 per pack for Drug X instead of GBP 740, a price at which the ICER is GBP 28,750 per QALY, above the threshold (computed here for illustration). Savings enter Delta_C_other as negative amounts and raise the EJP; added costs lower it.

• Mixing per-patient and per-unit EJPs

  Drug X's per-patient EJP of GBP 18,500 and its EJP of GBP 740 per pack describe the same ceiling. Comparing the per-patient figure with a pack price, or dividing by a unit count that differs from the one in the model, misstates the price: over 30 packs with no extra QALYs the EJP per pack falls to about GBP 616.67.

• Reading the EJP as a target price

  At the EJP the health gained is just offset by the health displaced elsewhere, so the net health benefit to the payer is zero and the value of the gain goes to the manufacturer as revenue, as Claxton and colleagues and the York glossary state. The EJP is a ceiling for the net price, NICE does not use a precise maximum acceptable ICER (PMG36 section 6.3.1), and meeting the EJP does not guarantee a positive recommendation.

Sources

• York glossary definition of the economically justifiable price

  York Health Economics Consortium. Economically justifiable price [glossary entry]. York: YHEC; 2016, updated October 2025. Defines the EJP as the maximum price at which an intervention would still be an efficient use of limited resources, often estimated as the price at which the ICER equals, or is just below, the threshold; at that price the net health benefit would conceptually be zero, and the EJP is a benchmark that does not dictate the pricing decision.

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• Claxton on the price at which the ICER 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. Section on the principles of value based pricing: a price at which the ICER just equals the threshold ensures that the health benefits of the drug are just offset by the health displaced elsewhere in the NHS, so the net health benefits to the NHS are zero and all the benefits of the innovation go to the manufacturer as revenue.

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• Mladsi on the zero-price conundrum

  Mladsi D, Barnett CL, Mader G, Russell-Smith TA, Unuigbe A, Bell T. The zero-price conundrum: exploration of scenarios where a clinically effective new drug might not be cost-effective at zero price. Value in Health. 2023;26(3):384-391. Abstract: a clinically effective drug may justify no more than a price of zero on cost-effectiveness grounds, mainly because of costs beyond its influence such as background disease costs, combination partner drugs and future interventions.

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• NICE reference case on discounting and net prices for the EJP

  National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; published 31 January 2022, last updated 31 March 2026. Section 4.4.4: reference-case analyses use prices that reflect as closely as possible the prices paid in the NHS, including patient access schemes and commercial access agreements. Section 4.5.1: costs and health effects are discounted at the same rate of 3.5% a year. Section 6.3.1, in the committee recommendations chapter: the committee does not use a precise maximum acceptable ICER.

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