Eur J Health Econ DOI 10.1007/s10198-014-0626-0

ORIGINAL PAPER

Using price–volume agreements to manage pharmaceutical leakage and off-label promotion Hui Zhang • Gregory S. Zaric

Received: 27 June 2013 / Accepted: 5 August 2014  Springer-Verlag Berlin Heidelberg 2014

Abstract Unapproved or ‘‘off-label’’ uses of prescription drugs are quite common. The extent of this use may be influenced by the promotional efforts of manufacturers. This paper investigates how a manufacturer makes promotional decisions in the presence of a price–volume agreement. We developed an optimization model in which the manufacturer maximizes its expected profit by choosing the level of marketing effort to promote uses for different indications. We considered several ways a volume threshold is determined. We also compared models in which offlabel uses are reimbursed and those in which they are forbidden to illustrate the impact of off-label promotion on the optimal decisions and on the decision maker’s performance. We found that the payer chooses a threshold which may be the same as the manufacturer’s optimal decision. We also found that the manufacturer not only considers the promotional cost in promoting off-label uses but also considers the health benefit of off-label uses. In some situations, using a price–volume agreement to control leakage may be a better idea than simply preventing leakage without using the agreement, from a social welfare perspective. Keywords Leakage  Pharmaceutical promotions  Risk sharing  Price–volume agreement  Off-label JEL Classification

C6  C7

H. Zhang (&) Faculty of Business Administration, Lakehead University, Thunder Bay, ON P7B 5E1, Canada e-mail: [email protected] G. S. Zaric Richard Ivey School of Business, Western University, London, ON N6A 3K7, Canada e-mail: [email protected]

Introduction In many jurisdictions, multiple requirements—such as safety, efficacy, and quality—must be satisfied before a drug is granted marketing authorization. Besides these three requirements, many third-party payers also use a ‘‘fourth hurdle’’, cost-effectiveness analysis, to decide whether a drug is included on a formulary [1]. Formularies may impose additional ‘‘limited use conditions’’ (LUCs) to restrict categories of use of listed drugs. For example, one LUC might stipulate that the drug can only be prescribed to patients who have experienced toxicity or treatment failure with a less expensive drug. We use the term ‘‘labeled’’ to indicate uses with regulatory approval, ‘‘off-label’’ to indicate uses which have not been approved, and ‘‘listed’’ to indicate acceptance on a formulary by a payer. In practice, a drug with the same chemical ingredients may have several indications: some of them are labeled and listed on a formulary; some are labeled but not listed; and some are off-label. The drug manufacturer may seek for approval for some indications by providing evidence on their safety, efficacy and quality. However, due to limited market size and expensive applications the manufacturer might not apply for their approval even though they have evidence of safety and effectiveness for certain diseases. In many cases, a drug listed on a formulary may be categorized as ‘‘general benefit’’ or LUCs. A drug under general benefit will be reimbursed by the payer for all possible uses. However, under LUCs, drug indications that are labeled but not cost-effective are usually not covered by a payer and are paid out of pocket by patients at a price charged by the manufacturer. The Ontario Drug Benefit in Canada also has an Exceptional Access Program (EAP), in which a physician can write a special request for the coverage of certain critical uses that are not listed but have no

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better listed alternatives. Thus, a drug with multiple indications may be covered by a payer for various uses including labeled and listed uses, unlisted uses and offlabel uses. If health and safety requirements are satisfied, then physicians are often able to prescribe approved drugs ‘‘offlabel’’ for any indication they believe is effective, even without regulatory approval for that specific indication. Off-label use is quite common and legal in many countries—for example, Radley et al. [2] found that off-label uses in the US account for approximately 21 % of prescriptions and that most of the off-label uses (73 %) may not be effective. Tabarrok [3] found that approximately 60 % of cancer patients, 80–90 % of pediatric patients, and 80 % of AIDS patients had used off-label drugs. Some State and Federal laws in the US even require payers to cover off-label uses. In Canada, off-label uses may be covered by the provincial health plans. Because of separate pricing regulation, drug manufacturers are not allowed to charge a different price for the off-label uses of the same drug. ‘‘Leakage’’, defined by Coyle et al. [4] is non-adherence to LUCs imposed by formularies, and is unavoidable for some drug indications. Leakage results in approved and listed drugs, which presumably have positive net benefit in one group or for one indication, being used for another group or indication where the net benefit is potentially negative or uncertain. We extended the scope of the analysis of leakage by Coyle et al. [4] to include off-label uses. When a drug is listed on a formulary under LUCs, leakage can occur from cost-effective indications to non-costeffective indications and from labeled indications to offlabel indications. Promotion of labeled uses is generally allowed, while promotion of off-label indications is generally illegal. In the US, drug manufacturers are prohibited from promoting off-label uses to physicians, but they can provide medical publications to physicians. Despite the presence of regulations prohibiting off-label prescriptions, there have been many high-profile instances of drug manufacturers being fined for this activity (e.g., [5–13]). There are controversial arguments about the value of off-label promotions. Some believe that truthful and non-misleading discussions between sales representatives and physicians are informative and beneficial to patients’ health. Others suggest that off-label promotion is often misleading and that, by allowing off-label promotion, drug manufacturers will have no motivation to seek FDA approval of these indications [9]. The manufacturer’s promotional activities, either legal or illegal, are usually designed for certain indications. However, promotion for one indication may increase not only the demand for that indication, but also demand for

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the undesignated indications. That is, the leakage from one indication to another is exacerbated by promotional activities. Leakage caused by promotional activity is related to the general marketing concepts of ‘‘spillover’’ and ‘‘brand extension.’’ Sullivan [14] defined ‘‘spillover’’ as follows: ‘‘Spillover occurs when information about one product affects the demand for other products with the same brand name.’’ Swaminathan et al. [15] defined ‘‘brand extension’’ as ‘‘to attach an existing brand name to a new product introduced in a different product category.’’ These concepts apply to drugs in that the listed uses, approved but unlisted uses, and off-label uses can represent a distinct product. Although promotion on off-label uses is illegal, violations occur often. The US federal government has pursued many cases against drug manufacturers and their employees on promoting off-label uses. In 2009, Pfizer paid 2.3 billion dollars, the largest penalty for health care fraud in the US, to settle the allegation for its illegal marketing of painkiller and other drugs [12]. This was Pfizer’s fourth settlement over illegal marketing since 2002. Other pharmaceutical companies have also been involved in similar cases. In 2013, Pfizer paid $491 million for the allegation of its off-label marketing on a kidney transplant drug [13]. An argument used against prohibitions on off-label marketing is that they violate ‘‘free speech’’ under the First Amendment in the US Constitution. Price–volume agreements have been proposed to address problems of leakage and spillover in prescription drugs [16–18]. A typical price–volume agreement works as follows: the payer and the manufacturer agree on a volume threshold. If the total sales volume exceeds the threshold, then the manufacturer rebates a portion of sales in excess of the threshold to the payer. By setting an appropriate volume threshold, the decision maker is able to control the total sales of the drug and thus limit off-label uses. An unintended consequence of controlling off-label uses may be a reduction in sales for cost-effective uses in the listed markets. In this paper, we aimed to answer the following research questions. How does a price–volume agreement influence the promotional decisions of a drug manufacturer? How should the volume threshold be determined and by whom? Can a payer use a price–volume agreement to prevent offlabel uses while enhancing cost-effective uses? What is the impact of leakage and spillover on decisions by the manufacturer and the payer? To answer these questions, we first developed a general model in which off-label promotion occurs and a payer tries to use a price–volume agreement to limit the amount of leakage. The manufacturer can exert promotional effort to increase sales, and this effort can be directed to specific indications, which may be labeled or off-label. Although promotional efforts can be directed to specific indications, spillover effects cause leakage to other indications. Later, we investigated a special case in which

Price–volume agreements to manage pharmaceutical leakage and off-label promotion

off-label promotion can be prevented. We then identified the impact of preventing off-label promotion on the optimal decisions and both parties’ performance.

Literature review Price–volume agreements have been widely used in the UK, Australia, New Zealand, Italy, France, Spain, and Canada [19–21]. Although price–volume agreements are negotiated between the drug manufacturer and the payer (i.e., the formulary), the reimbursement price might not be part of the negotiation. The price of the drug may already be set if the current formulary negotiation represents a new use for an existing product (e.g., a new indication for an existing oncology drug). Also, in some jurisdictions a separate regulator is involved in pricing decisions. For example, in Canada the Patented Medicine Prices Review Board (PMPRB) regulates the brand name drug prices to make sure it is not ‘‘excessive’’ [22]. Each province may engage in further price regulation. For example, British Columbia uses reference prices, which define the reimbursement price as the price of the most cost-effective alternative in the same therapeutic class [22]. In Australia, there are several pricing mechanisms for the payer to choose when determining the formulary price. One pricing mechanism is to calculate the weighted average of all prices based on their relative uses. Another mechanism is to take the weighted average of monthly treatment cost for all drugs in the same therapeutic groups based on their relative volume of use [23]. We thus assumed that the reimbursement price is regulated and calculated based on a certain pricing mechanism and thus is exogenous in this paper. There are many studies on the advertising spillover effect between two products under the same brand name. Sullivan [14] found that the introduction of a new car model increases demand for existing models, while advertising on the new model decreases demand for existing models. This mixed spillover effect may be caused by the substitution effect between the two models. Thus, Balachander and Ghose [24] separated the substitution and spillover effect and found that advertising on the extended products for yogurt and detergent brands has a positive impact on the demand for their old products. However, advertising on the old products does not increase demand for the extended products. Swaminathan et al. [15] found that successful brand extensions have positive reciprocal spillover effects while failed extensions may potentially have negative spillover effects. Erdem and Sun [25] examined the spillover effect of advertising in toothpaste and toothbrush markets and found that advertising in each market has a positive effect on the other market. Lei et al.

[26] showed that the spillover effect between brands depends not only on the strength of brand association but also on the direction of the association. Since we were not aware of any studies on spillover and extension for drugs, we assumed the effects of marketing would be similar to those just described for other consumer goods. Thus, in this paper, we assumed that there are mutual spillover effects between promotions on different indications. Numerous studies have documented the impact of promotions on the drug’s market size as well as on the price sensitivity of demand. They found that promotions not only increase the market size of a drug, but also may reduce the sensitivity to the price. For instance, Mizik and Jacobson [27] found that detailing and free samples have positive and significant effects on drug prescriptions. Similar results are obtained by Rizzo [28] for antihypertensive drugs in the US and Donohue and Berndt [29] in the antidepressant market. There are also some examples on the impact of promotion on the price sensitivity of pharmaceutical products. Rizzo [28] empirically tested that advertising decreases price elasticity of demand. Gonul et al. [30] investigated the influence of pricing and promotion on physicians’ prescription behavior and found that physicians whose patients are covered by Medicare or HMO are quite price-insensitive. Windmeijer et al. [31] used data for drugs from 11 therapeutic markets and found that drug promotion reduces the already small price sensitivity to almost zero. We thus assumed that promotion increases demand, which is price insensitive because patients are covered by the third party payer and the promotion will reduce the price sensitivity to zero. We were aware of very few modeling papers that investigate the prevention of leakage in the pharmaceutical market. Coyle et al. [4] proposed using stratified net benefits to determine optimal LUCs. They assumed that leakage increases total drug use and may reduce the average cost-effectiveness of reimbursed drugs if there is substantial leakage to non-cost-effective indications. They assumed a constant rate of leakage across use categories and did not consider factors that might change the rates of leakage. We extended this analysis in three ways: (1) by including leakage not only from cost-effective uses to noncost-effective uses, but also from labeled uses to off-label uses; (2) by assuming that the amount of leakage is not constant but is, instead, influenced by the level of promotional activity by the drug manufacturer; and (3) by assuming that, in addition to using LUCs, the payer uses a price–volume agreement in an attempt to control the amount of leakage to off-label or unlisted indications. Although price–volume agreements have been widely used, we were aware of few attempts to model this type of contract. Zaric and O’Brien [32] analyzed the optimal threshold decision for a manufacturer presented with a risk-

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A drug submitted to the regulator for certain indication(s)

Table 1 Summary of notation Comments Decision variables

Indication(s) approved and labeled

Indication(s) costeffective and listed on a formulary Market 1

Other indications not approved and thus off-label Market 3

T

Volume threshold

mi

Effect of promotional efforts on demand in market i

Objectives and constraints E½p

The manufacturer’s expected profit

D

The weighted difference between demand in the primary market and the secondary markets

Indication(s) not cost-effective and not listed

NMB Net monetary benefit of all drug uses Random variables

Market 2

ei

The random component of demand in market i

e

The random component of total demand, e ¼ e1 þ e2 þ e3

Fig. 1 Description of thes three markets

Ni

Parameters Qi

sharing agreement. Zhang et al. [33] used a game-theory perspective to investigate the optimal design of a price– volume agreement from a payer’s perspective. However, neither article explicitly considered promotional effort or leakage by manufacturers. Zhang and Zaric [34] developed a leakage model in which the drug manufacturer determines its marketing effort in labeled and off-label markets and the promotional effort in one market may increase sales in other markets. They considered only the case where the volume threshold of the price–volume agreement is exogenously determined. We extended their work by including the decision of the volume threshold, determined either by the drug manufacturer or by the payer, and by taking a game-theory perspective to solve the problem. Other researchers have addressed uncertainty in health benefits as opposed to sales [35, 36].

Model structure and assumptions We built on the framework described by Zhang and Zaric [34]. We considered a drug that is included on a formulary under LUCs because it is only cost-effective for certain indications. We categorized the drug’s indications into three markets (Fig. 1). We referred to the first as the primary market (i = 1), in which all drug indications are costeffective. We referred to the other markets as secondary markets. These include unlisted uses that are not costeffective, and thus not covered by a formulary (i = 2), and off-label uses that are not approved by the regulatory authority and that may or may not be cost-effective (i = 3). We assumed that a formulary would only list a product which had received regulatory approval for at least one indication that proves to be cost-effective. Sales in either of

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Demand in market i

Baseline demand in market i

FðeÞ; f ðeÞ

c.d.f. and p.d.f. of e

hi

The health benefit of the drug per person in market i

p

Unit price of the drug

c

Marginal production and distribution cost

a

Rebate rate

lij

Leakage factor from market i to market j

ui ðmi Þ

The marketing cost for marketing effort in market i

Calculated marketing effect and threshold mE; i

The optimal marketing effect in market i for (E)

mE;u i

The optimal marketing effect in market i for (E) without constraints

mM; i , T M;

The optimal marketing effect and threshold in market i for (M)

T M;N

The largest solution of E½NMBðTÞ ¼ 0 given m ¼ mE; i

mM;N i

The optimal solution to (E) when T ¼ T M;N

P; mP; i ;T

The optimal marketing effect and threshold in market i for (P)

T P;I ; T P;D

The smallest and largest positive solution of gP ðTÞ ¼ 0

P;D mP;I i ; mi i

The optimal solution to (E) when T ¼ T P;I or T ¼ T P;D

T

The value of T beyond which mE; ¼0 i

T; T

T ¼ minðT 1 ; T 2 ; T 3 Þ, T ¼ maxðT 1 ; T 2 ; T 3 Þ

the secondary markets represent leakage. The payer would cover most of the expense for the primary use. The payer may also cover some uses for the secondary uses upon special requests by physicians or may be obliged to pay for off-label uses if physicians think necessary. In this paper, we focused on the situation in which the payer covers all expenses incurred in all three markets. All notation is summarized in Table 1. We assumed that the unit price of the drug, p, is exogenous and is the same in all markets. For example, in Ontario Canada, a form of price–volume agreement is negotiated between the Ministry of Health and the drug

Price–volume agreements to manage pharmaceutical leakage and off-label promotion

manufacturer on brand name drugs seeking to be listed on the Ontario Drug Benefit program [22]. The payer will pay a reimbursement price for all claims under a volume threshold and pay a reduced price for claims above the threshold. Let di be the average units of consumption of the drug and bi (measured by quality adjusted life years) be the average incremental health benefit per person in market i.1 Let si be the incremental cost per person associated with the drug, excluding the drug price. Let k be the payer’s willingness to pay (WTP) per unit of bi. The net monetary benefit (NMB) of the drug per person in market i is NMBi = kbi - si - dip, and the average NMB per unit is NMB ¼ ðkbi  si Þ=di  p: To simplify the notation, we defined hi = (kbi - si)/di as the net benefit excluding drug costs. Thus, by definition of the three markets, h1  p  0; h2  p\0 and h3  p may be positive or negative. Let mi be the additional demand (in units of drug) in market i caused by the promotional effort targeted at market i. Although off-label promotion is generally illegal, as described earlier in this paper, it does occur (e.g., [5– 13]) and it is difficult and expensive to prevent. We thus initially described and solved a general model to represent the real world situation in which off-label promotion can occur (i.e., m3 can be positive), and we later considered the impact of restrictions on off-label promotion (i.e., adding a constraint of the form m3 = 0). Let ui ðmi Þ be the cost of achieving additional demand level mi in market i. We assumed that ui ðmi Þ is quadratic in 0 mi with ui ðmi Þ [ 0 (marketing can only increase sales), 00 ui ðmi Þ [ 0 (there are diminishing marginal returns on marketing investment), and ui ð0Þ ¼ 0 (no cost occurs without promotional effort). We assumed that the promotional effort in market i increases demand in market i and also has a spillover effect in the other markets. Numerous studies have documented the association between marketing and increased overall sales in the pharmaceutical industry (e.g., [28, 29, 31, 37]). Positive or negative spillover effects are also found between consumer products and extended brands [15, 24, 25]. The spillover effect is captured by parameter lij defined as the ‘‘leakage factor’’ from market i to market j. The leakage factor lij determines the extent to which effort in market i increases demand in market j. In particular, the ‘‘leaked effect’’ in market j is lij mi (similar to the expression for the impact of advertising used elsewhere [38]). We 1

mi does shift Ni but in each of the three separate markets, and these three separate markets, by definition, have different values of hi. In this way the average benefit obtained by the payer does vary in Mi. This is an approximation to the more general case where total benefits vary by promotional effort [i.e., b(mi)], which we discuss in the conclusions.

assumed lii ¼ 1 and 0  lij  1, meaning that the leaked effect in market j is less than in the original market i. Let P Mi ¼ 3j¼1 lji mj be the total marketing effect in market i. Let Ni be the demand (total units of the drug) in market i, defined as Ni ¼ Qi þ Mi þ ei , where Qi is the baseline (deterministic) demand in market i without marketing effort (i.e., Qi represents sales in market i when Mi is 0) and ei is a stochastic component representing demand uncertainty with E[ei] = 0. We assumed that Qi is independent of the drug price for the following reasons. First, in our setup, the price–volume agreement is negotiated between the payer and the drug manufacturer. Thus, by definition, patient purchases are covered by the payer. Second, the prescription decision is usually made by physicians rather than patients, and physicians, especially those whose patients are covered, are price insensitive [30]. Third, for some drugs the price sensitivity can be reduced to almost zero by promotions [31]. Note that Bala and Bhardwaj [39] also assumed price independent demand in a model of direct-to-consumer advertising and detailing decisions of a drug manufacturer. The distribution of ei is known to both parties with pdf fi() and cdf Fi(). We defined the total stochastic component as e ¼ e1 þ e2 þ e3 ; having pdf f() and cdf F(). Let T be the threshold value specified in the price–volume agreement. The threshold is negotiated between the payer and the manufacturer. However, it is unclear how the negotiation takes place. It may be estimated based on the manufacturer’s stated sales estimate [22]. However, since the forecast often can be overstated and is hard to monitor, we also considered the case where the payer may estimate the threshold and specify it in the agreement, thus making T a decision by the payer. The price of units sold exceeding T is reduced by a ratio a, a C 0, of the original price. We assumed that a is exogenous. Publicly available details on several risk sharing plans suggest that a rebate of 100 % is common [40]. Thus, the price is reduced to (1 - a)p if a B 1. Otherwise, the manufacturer returns not only all excess sales but also a portion (a - 1)T to the payer as an extra penalty. The reduction in price is equivalent to a rebate of R ¼ maxf0; aðN1 þ N2 þ N3  TÞg from the drug manufacturer to the payer. The manufacturer aims to maximize expected profit E½p; which includes the contribution generated from all three markets less manufacturing costs, promotional expenses, and rebates. We assumed that the marginal production and distribution cost is c per unit, with 0 B c \ p. We ignored research and development costs because they are sunk costs at the time of formulary negotiation. The manufacturer’s profit for the three markets P P is p ¼ ðp  cÞ 3i¼1 Ni 3i¼1 ui ðmi Þ  apR: Thus, the expected profit is

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H. Zhang, G. S. Zaric

E½p ¼ ðp  cÞ

3 X

" E½Ni   apE

i¼1



3 X

3 X

!þ #

¼ 0 for T  T i (iii) mE; i

Ni  T

E;

 (iv) dE½p dT  0: (v) E[NMB] either increases or decreases with T.

i¼1

ui ðmi Þ:

ð1Þ

i¼1

The overall expected NMB of the drug is 3 X ðhi  pÞE½Ni  E½NMB ¼ i¼1   þ apE ðN1 þ N2 þ N3  TÞþ :

ð2Þ

In the next three sections, we consider three variants of this problem, depending on whether the threshold T is determined exogenously, is chosen by the manufacturer, or is chosen by the payer.

Models and analysis Base model (E) We first briefly describe the base case model [which we refer to as (E)] in which the threshold T is exogenous in the price–volume agreement before the manufacturer makes its promotional decision. We considered a generalization of the model presented earlier [34] in which 0 B c \ p (in the previous model only c = 0 was considered). Let ui ¼ 0 0 00 00 ui ðmi Þ; such that ui ¼ ui ðmi Þ and ui ¼ ui ðmi Þ: To simplify the notation let F  F ðT  E½N1 þ N2 þ N3 Þ and f  f ðT  E½N1 þ N2 þ N3 Þ: The manufacturer solves the following profit maximization problem in (E): ðEÞ max E½p ð3Þ m1 ;m2 ;m3

s:t: mi  0;

i ¼ 1; 2; 3:

mE;u i

Let denote the unconstrained solution of (E) (i.e., the solution to maxE½p by ignoring the non-negative constraints) so that mE;u is obtained by solving i oE½p=omi ¼ 0; i.e., 3 X 0 lij  ui ¼ 0; i ¼ 1; 2; 3: ð4Þ ðpð1  að1  FÞÞ  cÞ j¼1

Thus, the optimal promotional effect in market i is   mE; ¼ max 0; mE;u : Let T i be the value of T that solves i i ¼ 0; i ¼ 1; 2; 3: mE;u i Proposition 1 problem (E). dmE; i

omiE;

The manufacturer chooses the threshold level (M) In this section, we developed a model in which the manufacturer determines not only promotional effort but also the volume threshold T. The sequence of events is shown in Fig. 2a. We assumed that the manufacturer solves for T and reports it to the payer. Based on the reported T, the payer calculates E[NMB] by anticipating the manufacturer’s optimal promotional decisions. The payer may reject listing the drug on the formulary (in which case profit is zero) if it believes that the overall NMB would be negative. The manufacturer thus includes an NMB constraint when solving for T to reflect decision-making by the payer. Once the drug is listed on the formulary, the manufacturer determines the optimal promotional effort, as shown in problem (E). This problem is solved in reverse time sequence. First, the manufacturer solves (E). Next, the payer decides whether to accept the agreement depending on the expected NMB. Given mE; and the payer’s E[NMB], the manufaci turer solves the following profit maximization problem: ðM Þ

The following properties are true for omiE;

(i) dT  0; oc  0 and oa  0: omE; (ii) opi may be positive when a is relatively low and negative otherwise, i = 1, 2, 3.

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The proof is similar to those provided in Zhang and Zaric [34] except that c [ 0 in this paper. It is intuitive that a larger threshold motivates the manufacturer to promote more and the price–volume agreement helps reduce the promotional effort [case (i)]. However, it is unclear how the price will influence the promotion decision. If the rebate is relatively low (a \ 1), then a higher price will incent more promotion. However, this is not necessarily true if the rebate is sufficiently high (a [ 1), as shown in case (ii). Thus, a high penalty helps the payer control promotional effort, and it is especially effective for an expensive drug. Case (ii) shows the situation in which promotion is completely prevented by setting a sufficiently low threshold. This situation, however, may not occur if the threshold is determined by the payer or the manufacturer, as discussed in the remaining sections. It is obvious that the manufacturer can benefit from a higher threshold [case (iv)] but surprisingly the payer may also be better off in terms of NMB [case (iv)].

max E½pðmE; i Þ T

s:t: E½NMBðmE; i Þ  0

ð5Þ ð6Þ

T  0: Given mE; i ; as a function of T, E[NMB] becomes a function of T, denoted by gM ðTÞ:

Price–volume agreements to manage pharmaceutical leakage and off-label promotion Fig. 2 Sequence of events. a Sequence of events for problem (M), b sequence of events for problem (P)

a Sequence of events for problem (M)

b Sequence of events for problem (P)

M

g ðTÞ ¼

3 X

ðhi  pÞ Qi þ

i¼1

"

þ apE

3 X

!

(ii)

mE; j lji

j¼1 3 X i¼1

Qi þ

3 X

! mE; j lji

þ ei

!þ # T

:

j¼1

ð7Þ M

There may be multiple values of T that satisfy g ðTÞ ¼ 0: Let T M;N denote the largest value of T that solves gM ðTÞ ¼ 0: Let mM;N denote the optimal solution to (E) i when T ¼ T M;N . The solution to (M) is stated as follows. Solution to (M) (i) (ii)

¼ mE; If gM ð1Þ  0 then T M; ¼ 1; mM; i i jT¼1 and E½NMB  0: Otherwise, E[NMB] = 0. There are two sub-cases. (a)

(b)

If T M;N  0 then T M; ¼ T M;N and mM; ¼ i mM;N : E[NMB] decreases with T at i M; M;N ¼T : T If T M;N \0 then there is no feasible solution.

Case (i) illustrates an extreme situation in which the manufacturer sets a very high threshold to avoid any rebate and the payer can obtain positive NMB. This occurs only when the health benefits are relatively high compared to the drug cost so that the NMB constraint can be easily satisfied even without a rebate. In contrast, when the health benefits are relatively low, it is impossible to obtain a positive NMB without the assistance of a price–volume agreement. Thus, the manufacturer has to choose a moderate threshold to meet the payer’s requirement. If the health benefits are sufficiently low, the manufacturer cannot meet the NMB constraint no matter how low the threshold value is set. For this case, no agreement can be reached. Proposition 2 summarizes the properties of the optimal solution to problem (M). Proposition 2 (i)

dT M;N dh i

M;N

M;N

M;N

 0; dTdp  0; dTda  0 and dTdc ¼ 0:

dmM;N dh i

M;N

 0 and omoc  0; and mM;N may be increas-

ing or decreasing with respect to p and a. (iii)

omiE; jT¼1 ohi omiE; jT¼1 oc

¼ 0;

omiE; jT¼1 op

 0;

omiE; jT¼1 oa

¼0

and

 0:

It may be expected that manufacturers have less motivation to promote higher quality products. A good quality drug may imply a high base demand, which discourages the manufacturer to promote if T is exogenously given. However, if the manufacturer sets the threshold, it tends to set a higher threshold and thus is able to promote more. This happens when higher health benefits help the payer achieve a higher positive NMB and the drug manufacturer can promote to maximize its profit without worrying about the NMB constraint. When the health benefits are increased to a certain level so that a positive NMB is guaranteed, the manufacturer will promote more as the drug price rises. The third term in case (i) shows that a higher penalty for excess sales will boost the threshold. It may seem intuitive that more promotional effort will be exerted by the manufacturer when confronted with a higher rebate. However, this is not the case, as seen in case (ii). This is because of two offsetting effects: on the one hand, a high penalty results in a high threshold and thus high promotional effort because the payer benefits from receiving a large return; on the other hand, it discourages the manufacturer from promoting because the manufacturer is worse off because of paying a large rebate. Thus, it is difficult to judge the impact of the price–volume agreement on the optimal promotional effort in problem (M), while in (E) the price– volume agreement helps control promotions. The payer chooses the threshold level (P) In this case, we considered the situation under which the payer will choose T. We assumed that the payer wishes to promote cost-effective uses in the primary market and to control or prevent undesirable uses in the secondary markets. To achieve this, the payer defines a weight j

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representing the importance of sales in the secondary markets versus sales in the primary market. The payer then aims to maximize the weighted difference between expected sales for listed indications and for leaked indications, given by D ¼ E½N1   jE½N2 þ N3 : A high value for D is beneficial to the payer and represents mostly appropriate (listed) uses, while a low or negative value of D represents mostly undesirable or inappropriate uses. If j [ 1; then the payer is more concerned about reducing leakage than increasing demand in the primary market. As j ! 1; the payer’s objective shifts towards minimizing leakage. A special case is j ¼ 0; in which the payer aims to maximize use of listed cost-effective drugs. Although the payer may set different weights for the demand in markets 2 and 3 to reflect different goals, we assumed that the weights for the secondary markets are the same without losing the generality of the problem. When j\0 the payer’s objective D becomes the weighted total demand of all three markets.2 The payer sets the threshold to maximize D in anticipation of the manufacturer’s promotional decisions. This is the major difference from problem (M), in which T is chosen by the manufacturer to maximize its profit. The sequence of events is shown in Fig. 2b. The payer solves its problem by anticipating the future promotional effort E; mE; i : Given mi ; the problem becomes    E; E;  ðPÞ max D ¼ E N1 ðmE; i Þ  jE N2 ðmi Þ þ N3 ðmi Þ T

ð8Þ s:t: E



 NMBðmE; i Þ 0

T 0 There are several cases for the solution to this problem. 1. When T  T ¼ maxðT 1 ; T 2 ; T 3 Þ, D is linear in T. Thus, dD/dT [ 0 if j\j0 , dD/dT \ 0 if j [ j0 and dD/dT = 0 if j ¼ j0 , where P3 P3 00 00 P3 00 00 00 00 u2 u3 l þu1 u3 l21 l þu1 u2 l31 l j¼1 1j j¼1 2j j¼1 3j P3 P3 P3 j0 ¼ : 00 00 00 00 00 00 ðl12 þl13 Þu2 u3

l þð1þl23 Þu1 u3 j¼1 1j

l þð1þl32 Þu1 u2 j¼1 2j

l j¼1 3j

Without the NMB constraint, the payer will set T ¼ 1 when j\j0 and T = 0 otherwise. (2) When 0  T  T ¼ minðT 1 ; T 2 ; T 3 Þ; mi ¼ 0 for all markets. Thus, D is independent of T so that T can take any value in this range and E[NMB] decreases with T. (3) When maxð0; TÞ\T\T; mi ¼ 0 in one or two of the three markets. For this case, D will be increasing with T for relative small values of j while decreasing with T for relatively large values of j. The switching point of j from an increasing D to a decreasing D may differ for different 2 This is a special case of D being the total health benefit for all possible uses, in which the weights of the demand are the health benefits while in the special case the weights are 1 and j:

123

ranges of values of T. The largest D will be obtained either at T = 0, ?, or Ti, depending on the specific value of j. The optimal solution involves the consideration of many cases. Below, we present a special case when lij has the same value for all i, j = 1, 2, 3, in which the switching point of jwill be independent of T. Thus, D is increasing or decreasing with T for the same values of j regardless of the value of T. We then conduct numerical examples to illustrate some observations for this special case. Let gP ðTÞ denote the value of E[NMB] given mE; as a i function of T. Thus, gP ðTÞ is given in (7). Let T P;I  0 be the smallest positive solution to gP ðTÞ ¼ 0: When T\T; mi ¼ 0 for i = 1, 2, 3 and gP ðTÞ decreases with T. Thus, if gP ðTÞ  0; any T in the range of ½0; T satisfies E[NMB] C 0. Otherwise, T P;I is the smallest value of T that satisfies E[NMB] C 0, where dgP ðTÞ=dT\0 at T P;I : Let T P;D  0 be the largest positive solution to gP ðTÞ ¼ 0. Thus, if gP ð1Þ\0; then T P;D is the largest value of T that satisfies E[NMB] C 0 where d dgP ðTÞ=dT\0 at T P;D : Otherwise, E[NMB] C 0 for any T larger than T P;D : It is possible that T P;I ¼ T P;D if gP ðTÞ is monotonic in T. We used mP;I to denote the marketing effect mE; when T ¼ i i P;D P;I T and mi to denote the marketing effect when T ¼ T P;D : Thus, T P;D ¼ T M;N and mP;D ¼ mM;N : i i The solution to (P) is summarized below. Solution to problem (P) Case 1. j\j0 (i) If gP ð1Þ  0; then T P; ¼ 1 and mP; ¼ mE; i i jT¼1 : E½NMB  0: (ii) If gP ð1Þ\0; then T P; ¼ T P;D and mP; ¼ mP;D i i : E[NMB] = 0. Case 2. j  j0 (i) If gP ð0Þ  0; then T P; can be any value in ½0; maxð0; TÞ when gP ðTÞ  0 and can be any value in ½0; T P;I  otherwise. mP; ¼ 0 and E½NMB  0: i (ii) If gP ð0Þ\0; then T P; ¼ T P;I and dgP ðTÞ=dT [ 0 at T P;I if such a T P;I exists. mP; ¼ mP;I and E[NMB] = 0. i i When j is relatively small, the payer cares more about promoting listed demand [case 1]. For this case, the payer would like to set a high threshold value so that the manufacturer can promote aggressively. In particular, if the health benefit is relatively high, then the threshold value can be so high that no rebate will be imposed on any excess sales [case 1(i)]. If the health benefit is low, then the manufacturer may still need to return a portion of excess sales [case 1(ii)]. The health benefit in one market can compensate the health benefits in other markets. For instance, if the listed indications are very cost-effective, then the payer may still welcome aggressive promotion even though it is aware that the leaked indications are not cost-effective.

Price–volume agreements to manage pharmaceutical leakage and off-label promotion

When j is relatively large, the payer’s objective is to control leakage by setting a low threshold value [case 2]. In particular, the payer will set a relatively small threshold when the health benefit is relatively high, while setting a larger threshold when the health benefit is relatively low. This is the opposite of the case with a small j; in which the threshold value increases with the health benefit. In this case, we made several observations: (1) if the health benefits are relatively high, then the payer sets a threshold sufficiently low to prevent any possible promotion and the NMB thus decreases with the threshold [case 2(i)]. The payer would thus set the threshold as 0 if its goal is to achieve the highest NMB, and can set a higher threshold if its goal is just to prevent leakage. (2) If the health benefits are relatively low, then the payer would increase the threshold to allow promotion on less cost-effective drugs [case 2(ii)]. (3) With relatively large health benefits the payer can easily maintain a positive NMB and thus is able to reduce the threshold to a very low level. (4)With relatively small health benefits, reducing the threshold will discourage the manufacturer’s promotion and thus the payer may be unable to generate a positive NMB without enough demand in markets with positive net benefit. (5) If the health benefits are lower than the price in all markets, the NMB is always negative and thus it is impossible to reach an agreement between the two parties. Proposition 3 summarizes several properties of the optimal solution to (P). Proposition 3 (i) (ii) (iii) (iv)

(v)

(vi)

dT P; =dj  0 and dmP; i =dj  0: P;D P;I dT =dhi  0 and dmP;D i =dh  0; dT =dhi  0 P;I  and dmi dhi  0:  dT P;D =da  0 and dmP;D da is either positive or i  P;I da  0: negative; dT =da  0 and dmP;I i  dQ  0 if hi  p dT P;D =dQi  0 and dmP;D i i ð1  að1  FÞÞ and the contrary otherwise: it is opposite for T P;I and mP;I i :   P;D dlji  0 and dmP;D dlji  0 if hi  p dT i ð1  að1  FÞÞ and the contrary otherwise: it is opposite for T P;I and mP;I i : The optimal threshold is at either the lower bound or upper bound of the feasible solution.

Case (i) suggests that the payer would lower the threshold in order to reduce leakage, which may also discourage promotional activities in the primary market. As shown in case (ii), if the payer’s goal is to promote more cost-effective uses of drugs (j\j0 ), then it is willing to set a high threshold to encourage the manufacturer’s promotion when the health benefits are relatively high. However,

if the goal is to reduce leakage (j  j0 ), the payer will set a small threshold to reduce promotion when the health benefits are high. Given all other parameters, the decisionmaker may make opposite decisions given different objectives. To promote cost-effective uses of the drug, the payer is also willing to increase the threshold when the rebate rate is high [case (iii)]. However, the payer would not necessarily do so when either the base demand or leakage factor is high [cases (iv) and (v)]. It would do so only when the health benefits are sufficiently high. Example We created numerical examples to illustrate the properties of the optimal solution and the resulting performance for (P). We assumed that the cost of effort is quadratic to capture diminishing returns. In particular, we assumed uðmi Þ ¼ ai m2i =2 þ ki mi : All parameter values are shown in Table 2. Here, we set j ¼ 0:2: This combination of parameter values results in j0 ¼ 0:5 so that j\j0 : Figure 3 shows the optimal solution and the corresponding performance for both parties for various values of p. Figure 3a shows that the optimal promotional effort first increases and then declines with p. The sudden dip at p = 12 occurs when the E[NMB] changes from a positive value to zero so that the optimal promotional effort is generated from a different set of equations. When p C 16, there is no feasible solution because the payer is unable to find a T to satisfy E[NMB] C 0. Figure 3b shows that the optimal profit first increases and then decreases with p while the NMB declines with p. When p B 12, a higher price promotes the manufacturer’s profit and encourages more promotional effort because a positive E[NMB] is easily satisfied when the price is relatively low compared to the health benefits. In contrast, when p [ 12, a higher price may reduce the manufacturer’s profit and its promotional effort. A high price makes it difficult for the payer to satisfy the NMB constraint so that the payer has to set a low threshold to increase E[NMB], thus discouraging the manufacturer’s promotional effort. We also created an example to illustrate how a price– volume agreement influences the total social welfare, which we defined as the sum of the manufacturer’s profit and the payer’s NMB. First, we obtained the resulting social welfare when off-label promotion does not occur and no price–volume agreements are used (i.e., m3 = 0 and a = 0). Second, we changed a in case (M) given various values of p. Next, we compared the social welfare between the two steps to identify the impact of a price–volume agreement on off-label promotions. Figure 4 shows how social welfare varies with rebate rates at different prices. The solid lines represent the general case where off-label promotion occurs under a price–

123

H. Zhang, G. S. Zaric Table 2 Parameter values for the base case numerical examples Category

Parameters values in base case

Explanations

Baseline demand

Q1 ¼ 1; 500; Q2 ¼ 1; 000; Q3 ¼ 800

We assumed that baseline demand for listed indication is higher than for markets 2 and 3 because it is encouraged by the payer and thus is mostly used

Leakage factor

lij ¼ 1

We assumed the leakage factor as 1 to fully capture the effect of spillover. This factor will not change the conclusion and can be varied in a sensitivity analysis

Health benefit

h1 ¼ 15; h2 ¼ 10; h3 ¼ 10

Since market 1 is for cost-effective indication, we assumed it to be most costeffective, while market 3 may be more or less cost-effective

Unit price

p = 12

We assumed the price in the base case is lower than the health benefit in market 1 but higher than that in market 2 so that market 1 indication is cost-effective while market 2 is not cost-effective

Production and distribution costs

c=1

These costs are relatively low compared to drug price, and thus we set it to be a small value

Demand variance

r = 600

Demand variance is relatively high in pharmaceutical market and can be varied from case to case

Rebate rate

a=1

Rebate rate in Ontario, Canada, exceeds 1, including the full return of excessive sales and administration expenses

Marketing cost

a1 ¼ 0:1; a2 ¼ 0:2; a3 ¼ 0:3; k1 ¼ 1; k2 ¼ 2; k3 ¼ 3

We assumed that the marginal promotional cost in market 1 is lowest among the three markets because it is eligible and most accessible to both patients and physicians

uðmi Þ ¼ ai m2i =2 þ ki mi

Fig. 3 The optimal solution and performance as functions of p in (P). a The optimal marketing effort in each market. b The optimal profit and NMB E½pP;  and E½NMBP; 

(a) 350 300

m1

250 200 150

m2

100

m3

50 0 0

5

10

p

15

(b) 50000

40000

E[ ]

30000

20000

E[NMB] 10000

0 0

5

10

15

p

volume agreement. The dotted lines represent the special case where off-label promotion is prevented and no price– volume agreements are used. Figure 4 shows that when

123

p = 12, the social welfare is a constant over a. This is because the health benefits are relatively high compared to the price so that E[NMB] is always positive and the

Price–volume agreements to manage pharmaceutical leakage and off-label promotion

optimal threshold is infinity. This works like a situation where a price–volume agreement does not exist. That is, when the manufacturer chooses the threshold, it would try to set a high threshold so that it is able to promote off-label uses as much as possible if the health benefits are relatively high. This improves the social welfare compared to the situation where off-label promotion does not occur and no price–volume agreements are used. This is because allowing off-label promotion increases the manufacturer’s profit more than the loss of the payer’s NMB. Note that the threshold is infinity when health benefits are relatively high and thus the rebate does not make any difference between the two cases. Figure 4 also shows that when p = 14 the social welfare decreases with the rebate for the general case. This is mainly because the manufacturer’s profit is reduced by paying a higher rebate while E[NMB] = 0. No agreement is reached between the two parties when a \ 0.2 because the payer cannot obtain a positive NMB with such a low rebate. The social welfare is improved by allowing off-label promotion with a price–volume agreement for a relatively small rebate (a \ 0.5). Otherwise, the social welfare is higher when offlabel promotion is prevented and a price–volume agreement does not exist for a relatively large rebate. With relatively low health benefits, the payer has difficulty keeping NMB positive. Thus, E[NMB] = 0. If the rebate is low, then the payer should help the manufacturer by allowing off-label promotion while using a price–volume agreement to control it. If the rebate is high, the payer should try to forbid off-label promotion without using a price–volume agreement. We obtained similar conclusions for case (P). Comparison between (M) and (P) When the threshold is determined by the manufacturer, the manufacturer chooses both a threshold and a promotional effort to maximize the profit. When the threshold is determined by the payer, the manufacturer chooses only the promotional effort, and the payer chooses the optimal

threshold to maximize its objective while satisfying the NMB constraint. Since the two parties’ objectives are different, they may choose different threshold values. It is obvious that the manufacturer’s profit will be higher when it is able to choose the threshold than when the payer chooses it—and that the payer’s objective will be lower for the same reason. The impact on the threshold and promotional effort is less clear. Let DM; and DP; represent the value of D, given the optimal solution for problems (M) and (P), respectively. The comparison is formally summarized in the following proposition. Proposition 4 (M) and (P). (i) (ii) (iii)

The following are true when comparing

E½pM;   E½pP;  and DM;  DP; : If j\j0 ; then T P; ¼ T M; ; mP; ¼ mM; and i i E½NMBP;  ¼ E½NMBM;   0: M; If j  j0 ; then T P;  T M; and mP; i  mi : E½NMBP;  may be higher or lower than E½NMBM; :

It is obvious, as case (i) shows, that either the manufacturer or the payer benefits from being able to choose the threshold. When the payer’s main concern is to promote primary uses, it is willing to set the same threshold as the manufacturer’s optimal choice in problem (M), regardless of the relative value of the health benefit. Thus, the payer can ask the drug manufacturer to set a volume threshold if its objective is to promote primary uses—even though uses in the secondary markets are not cost-effective and achieve the same result as if the payer had chosen the threshold itself. When the payer aims to control leakage it will set a lower threshold than the manufacturer would choose no matter how high or low the health benefit in the secondary markets are. If the payer expects less cost-effective uses in the secondary market, then it would be better for the payer to choose a threshold value. However, for this situation, the payer is not necessarily able to achieve higher NMB than when the manufacturer chooses the threshold.

Fig. 4 The impact of the price– volume agreement on the social welfare for (M)

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H. Zhang, G. S. Zaric

Impact of off-label promotion

The impact of off-label promotion on (P)

Preventing off-label promotion is difficult and costly. We investigated how the price–volume agreement can help reduce off-label promotion under various situations in the last section. In this section, we investigated the conditions under which the manufacturer is prohibited from promoting off-label uses. Since we used mM; and mP; to denote i i the optimal marketing effect for problems (M) and (P), we then used mM;0 and mP;0 to denote the corresponding i i marketing effect when the off-label promotion effect is m3 = 0. Our focus was on the impact of off-label marketing on cases (M) and (P).

Based on the previous discussion, when j\j0 ; problems (P) and (M) have the same solution. Thus, the impact of preventing off-label promotion is the same for both cases. When j  j0 ; there are two sub-cases. If gP ð0Þ  0 (the health benefits are relatively high), then there is no promotional effort in any market, so preventing off-label promotion has no impact on promotion for labeled markets. If gP ð0Þ\0 (the health benefits are relatively low), then there is no feasible solution based on the parameter values given in Table 2 because gP ðTÞ is a decreasing function of T.

The impact of off-label promotion on (M) mM;0 is obtained from the constrained optimization probi lem in (M) by adding an additional constraint, m3 = 0. It is and mM; difficult to analytically compare mM;0 i i : We thus used numerical examples to identify the impact of off-label promotion on the optimal solutions and resulting performance. We used the parameter values shown in Table 2. Figure 5a shows the impact of off-label promotion on the optimal promotional effort for various values of h3. Preventing off-label promotion provides an incentive for the manufacturer to promote more in markets 1 and 2 when h3 is relatively low, while it does not change the effort level in markets 1 and 2 when h3 is high. The manufacturer tends to promote more and generate extra revenue from markets 1 and 2 to compensate for losses from off-label prevention. However, with a low h3, it has more difficulty meeting the NMB concern, thus setting a low threshold to balance the NMB concern while increasing the promotional effort in markets 1 and 2 to boost profit. As h3 rises, it tends to promote more and sets a higher threshold. When h3 is sufficiently high, the NMB constraint is satisfied, and its promotion focuses on profit maximization. The manufacturer then chooses to set a very large T to avoid any rebate without worrying about the NMB concern. For this case, its promotional effort in each market is independent of the total demand and, thus, preventing off-label promotion may impact the total demand but does not impact the decisions in other markets. It is obvious that, by having off-label promotion, the manufacturer is able to obtain more profit than without, as shown in Fig. 5b. However, it is not necessarily true that the payer obtains better NMB without off-label promotion. When h3 [ 9, the manufacturer chooses a T to maximize its profit while maintaining a positive NMB. However, when h3 = 10 (h3 - p \ 0), any promotion has a negative contribution to NMB. Thus, the payer obtains less NMB by allowing off-label promotion. As h3 rises, allowing offlabel promotion contributes to higher NMB.

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Conclusions and future directions We examined the promotional decisions of a drug manufacturer under different settings of negotiation over the volume threshold of a price–volume agreement. We considered situations where the volume threshold may be determined by either the manufacturer or by the payer. We also took a game-theory perspective and considered a situation in which the payer is concerned not only about the NMB but also with balancing sales in different markets. Controlling off-label promotion is costly and often ineffective. Our analysis considered whether a price- volume agreement, which is relatively easy to implement, could be used to control leakage and off-label promotion. Our analysis yielded a number of important insights. First, in many cases, T  ¼ 1; implying that the optimal price–volume agreement involves a single constant price as opposed to the price reduction of a typical price–volume agreement. Second, due to NMB concerns, setting a higher price does not necessarily incents the manufacturer to promote more. Third, if the payer aims to promote costeffective uses in the primary market, then its optimal decision is the same as the manufacturer’s choice. For this case, the payer can simply ask the manufacturer to choose the threshold value. However, if the payer aims to control demand in the secondary markets, then it should choose the threshold itself. Fourth, the price–volume agreement helps the payer control promotional effort if the threshold is exogenous, but this is not necessarily true if the threshold is determined either by the manufacturer or the payer. Fifth, if off-label promotion is strictly forbidden and this can be costlessly enforced, then it may increase labeled promotion but would have little impact on the promotional effort in each market. In particular, if the manufacturer chooses the threshold, then preventing off-label promotion may incent the manufacturer to promote more aggressively in the other markets; if the payer chooses the threshold and its goal is to control leakage, then off-label promotion can be automatically prevented by setting a low threshold. Finally, when

Price–volume agreements to manage pharmaceutical leakage and off-label promotion

(a) 0 0

5

10

15

20

-2

m 2M

-4

-6

-8

m 1M

h3

-10

(b) 2000 1600 1200 M

800

NMBM 400

TM 0 0 -400

5

10

15

20

h3

Fig. 5 The impact of off-label promotion on (M) as a function of h3 . a The incremental marketing effect associated with allowing off-label M; promotion in markets 1 and 2 (DmM  mM;0 i , i = 1, 2). b The i ¼ mi

incremental threshold, profit and NMB associated with allowing offM M; M;0 M M; M;0 label promotion  (DT ¼  T   T ,Dp ¼ E½p   E½p  and DNMBM ¼ E NMBM;  E NMBM;0 )

defining social welfare as the sum of both parties’ objective functions, in some cases using a price–volume agreement to control leakage may be a better idea than simply forbidding leakage. This may be a useful insight for regulatory authorities that are concerned with controlling leakage without violating free speech rules. Our analysis has limitations. We included the NMB constraint for the payer but did not include the participation constraint for the manufacturer. We assumed that the payer was concerned with NMB but not total budget. Future research could consider the impact of a budget constraint. We assumed that the price and the rebate rate were exogenous but, in some circumstances, they may also be decisions made by one or both parties in negotiating a price– volume agreement. It is unclear how the threshold value is determined in the price–volume agreement and other negotiation methods may be considered, such as a Nash bargaining game, similar to the approach taken by Antonanzas et al. [35]. The leakage factors were assumed to be known in this paper. However, they may be uncertain and

could thus be included in any negotiations. We took the health benefit bi as a constant for different uses of the drug. In reality, it may be influenced by the promotional effort. In future research, it would be interesting to consider the case where bi is a function of marketing effort or total market size [i.e., bi(mi) or bi(Ni)]. We only considered direct investment in promotional effort, but it may be possible to consider investment in the leakage factors as well. Since the health benefit may be influenced by the promotional effort, the model can be extended to investigate how promotional decisions will be made with respect to this concern. There may be multiple drug manufacturers and multiple payers for the provision of the same drug. We may consider this situation and extend the model in a future study. Acknowledgments This work was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) and a Premier’s Research Excellence Award. Earlier versions of this work were presented at POMS Annual Conference and INFORMS Annual Meeting.

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H. Zhang, G. S. Zaric

Appendix

d2 D d2 mE; d2 mE; 1 2 ¼ ð1  jl12  jl13 Þ þ ðl21  j  jl23 Þ 2 2 dT dT dT 2 d2 mE; 3 þ ðl31  jl32  jÞ : dT 2

Proof for proposition 2 By taking the derivative of E[NMB(T)] = 0 for both sides oE½NMB dT with respect to hi, we have oE½NMB oT dhi þ ohi ¼ 0: That . M;N oE½NMB is, dTdh ¼ oE½NMB [ 0 because oE½NMB \0 at ohi oT oT

dm1E;u

T M;N . By following the same logic, we obtain the properties of T M;N with respect to other parameters. dmM;N omE; oT M;N omE; dh ¼ oT M;N ohi þ ohi  0, and similarly we have

E;u Similar formulas are obtained for dmE;u 2 =dT and dm3 =dT; which are substituted into (10). Thus, when T  T ¼ maxðT 1 ; T 2 ; T 3 Þ; we have

M;N

i

i

omM;N oc

Note that dT

¼

00

00

00

u2 u3

P3

00 00 l þu1 u3 j¼1 1j

00

u2 u3 ð1þl12 þl13 Þ

P3

00 00 l þu1 u2 j¼1 2j

P3

l j¼1 3j



00

00

00

:

þu1 u2 u3 =apf

 0.

P3 P3 P3 00 00 00 00 00 00 dD ð1  jl12  jl13 Þ j¼1 l1j u2 u3 þ ðl21  j  jl23 Þ j¼1 l2j u1 u3 þ ðl31  jl32  jÞ j¼1 l3j u1 u2   ¼ :  00 00 P 00 00 P 00 00 P 00 00 00 dT u2 u3 3j¼1 l1j þ u1 u3 3j¼1 l2j þ u1 u2 3j¼1 l3j þ u1 u2 u3 apf

Proof to problem (P) Given mE; (which is a function of T), the payer’s objective i becomes a function of T: ! 3 3 3 X X X E; E; D ¼ Q1 þ li1 mi  j Qj þ lij mi : ð9Þ i¼1

j¼2

i¼1

The first-order and second-order derivatives of D with respect to T are

ð11Þ

The denominator of (11) is always positive because 00 ui [ 0: Thus, the sign of dD/dT depends on the sign of the numerator. Let C represent the numerator of (11), i.e., C ¼ P P 00 00 ð1  jl12  jl13 Þ 3j¼1 l1j u2 u3 þ ðl21  j  jl23 Þ 3j¼1 l2j P 00 00 00 00 u1 u3 þ ðl31  jl32  jÞ 3j¼1 l3j u1 u2 : C is a constant over T and is decreasing in j. Let j0 be the value of j where C ¼ 0: Thus,

P P 00 00 P 00 00 00 00 u2 u3 3j¼1 l1j þ u1 u3 l21 3j¼1 l2j þ u1 u2 l31 3j¼1 l3j j0 ¼ : 00 00 P 00 00 P 00 00 P ðl12 þ l13 Þu2 u3 3j¼1 l1j þ ð1 þ l23 Þu1 u3 3j¼1 l2j þ ð1 þ l32 Þu1 u2 3j¼1 l3j

dD dmE; dmE; ¼ ð1  jl12  jl13 Þ 1 þ ðl21  j  jl23 Þ 2 dT dT dT dmE; 3 ; þ ðl31  jl32  jÞ dT ð10Þ and

123

dD/dT [ 0 if j\j0 ; dD/dT \ 0 if j [ j0 and dD/dT = 0 if j ¼ j0 : That is, the payer would be indifferent about T if j ¼ j0 : Without the NMB constraint, the payer will set T ¼ 1 when j\j0 and T = 0 otherwise. Since 1 þ l23  l21 and 1 þ l32  l31 ; the second and third components

Price–volume agreements to manage pharmaceutical leakage and off-label promotion

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