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Proactive Planning of the Timing of a Partial Switch of a Prescription to Over-the-Counter Drug

Gila E. Fruchter Graduate School of Business Administration

Bar-Ilan University Ramat-Gan, 59000, Israel

Phone: 972-3-5318915 e-mail: fruchtg@mail.biu.ac.il

Murali K. Mantrala College of Business

University of Missouri-Columbia Columbia, MO 65211, USA

Phone: 573-884-2734 Fax no. 573-884-0368

email: mantralam@missouri.edu

Current version: November 10, 2009

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Proactive Planning of theTiming of a Partial Switch of a Prescription to Over-the-Counter Drug

Abstract

This paper focuses on proactive planning of a partial switch of a prescription (Rx) drug brand to over-the-

counter (OTC) status within the former’s patent-protected life. The planning issues are whether and when

to make this switch from the ‘single status’ Rx drug phase to the ‘dual status’ Rx+OTC phase with and

without the grant of a 3-year market exclusivity period for the OTC drug by the US Food & Drug

Administration (FDA). We formulate an optimal control problem that leads to a two-phase optimization

problem with finite-time horizon involving switching time-dependent switching cost and terminal value.

Applying a variational approach, we establish the conditions under which it can be optimal to make the

partial switch before the Rx drug patent expiry. We show that, counter to conventional thinking, the grant

of market exclusivity is neither necessary nor sufficient for a partial switch before the Rx drug’s patent

expiry. Further, we show in a functionally specified model-based illustration, involving varying

combinations of plausible ratios of OTC and Rx version margins and potential market sizes, that denial of

OTC market exclusivity can imply that the firm should advance rather than delay the partial switch, and

yield greater savings for Rx drug consumers. Implications for pharmaceutical Rx-OTC switch planners

and policymakers are discussed.

Key Words: Analytical Models, Optimal Control, Variational Approach, Pharmaceutical Marketing Strategy, Product Life Cycles, Product Management, Market Entry, Prescription and Over-the-Counter Drugs.

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1. Introduction

In the United States (US), branded prescription (Rx) drugs worth $63 billion are set to lose their

patents between 2007 and 2012, paving the way for generic copycat rivals to enter and take large chunks

of their business (e.g., Baltazar 2007). At the same time, very few new products are likely to emerge to

replace branded Rx drugs with expiring patents. As a result, many branded drug makers are now

considering Rx to over-the-counter (OTC) drug switch strategies as one way to extend their Rx product

lifecycles and revenues beyond the original Rx patent-protected periods (e.g., Mahecha 2006, Harrington

and Shepherd 2002, Parece, Tuttle, and Hector 2004).

Current US Food and Drug Administration (FDA) regulations allow for a complete switch (the

branded Rx drug is withdrawn when an FDA-approved OTC version of the basic molecule is introduced)

or a partial switch (from single to dual regulatory status), i.e., simultaneous marketing of Rx and OTC

versions of the same molecule albeit in different strengths with modified indications (Harrington and

Shepherd 2002). Furthermore, the 1984 US Drug Price Competition and Patent Term Restoration

(Waxman-Hatch) Act, provides an incentive to pharma makers to consider marketing less-expensive OTC

versions of their Rx brands that could lower consumers’ and insurers’ costs of prescription drug

treatment. Specifically, the FDA can grant the OTC version a market exclusivity period of three years

from the date of its approval if the Rx-to-OTC switch sponsors conduct new clinical investigations that

are essential to gain approval for OTC marketing. If not, the FDA can deny market exclusivity while

approving OTC marketing of the drug, as it did in the case of Pharmacia’s Rogaine hair regrowth

treatment, see, e.g., Hathaway and Manthei (2004).

Today, much industry attention is focused on the value of undertaking partial switches

proactively, i.e., when there is no immediate threat of generic rival entry (Evers 2004). A recent example

of a proactive partial Rx to OTC switch came in February 2007 when the FDA approved 60 mg orlistat

capsules as the first OTC weight loss aid for overweight adults under the brand-name Alli (Smith, 2007).

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However, Alli, launched by its maker Roche and marketing partner Glaxo Smith Kline (GSK) in June

2007, is nothing but a half-strength version of Roche’s Rx drug Xenical which continues on the Rx

market as a drug for obesity at higher doses. Roche’s original patent for orlistat was initially approved in

1999 with a June 2009 expiry date prior to which no generic Rx version of Xenical could enter the US

market. Thus, Roche–GSK introduced the OTC version Alli a little over two years before orlistat’s patent

expiry. Further, the FDA also granted GSK the 3–year OTC market exclusivity which expires in

February 2010 (refer, e.g., to DrugPatentWatch.com). Until then, the FDA cannot approve any

application for a generic version of Alli – effectively extending Roche-GSK’s monopoly of orlistat in the

OTC market by a year.

Other examples of proactive partial switches in the US include Lamisil (terbinafine

hydrochloride), Imodium (loperamide), and Pepcid (famotidine). In Lamisil’s case, its maker Novartis

successfully switched some of the indications, namely, athlete’s foot (tinea pedis) and body ringworm

(tinea corporis), to OTC status in 1999, reserving more serious indications like onchomycosis for Rx drug

treatment. At the time of this partial switch, Novartis still had six years’ Rx patent exclusivity remaining

on this lucrative topical product.

However, the wisdom and timing of proactive partial switches remain hotly debated issues in the

industry considering that gross margins for patent-protected Rx brands can be four to five times larger

than those of the OTC versions, e.g., King et al. (2000). Consequently, many traditional pharma Rx

brand managers believe it is best to “Never launch” a potentially cannibalizing, low-margin OTC version

of a successful Rx drug before the latter’s patent expires (Pfister 2004). For example, within six months

of Alli’s June 2007 launch, a physician survey by Decision Resources, Inc. revealed that half of the

respondents had prescribed Alli rather than Xenical Rx to their patients. However, advocates of proactive

partial switch planning e.g., switch.com’s Steve Francesco (2001), Datamonitor (2000), argue that with

additional market exclusivity, consumer loyalty can be transferred directly from the Rx brand to its OTC

version before generic rival entry, leading to a sizable ongoing OTC revenue stream. To take advantage of

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the OTC market exclusivity provision, these advocates typically recommend that the partial switch be

done about two years or less before Rx drug patent expiry. Clearly, however, firms like Novartis in the

case of Lamisil have followed different rules.

As the industry debate continues, a normative model to help a firm’s managers better understand

and resolve the “Whether to” and “When to” questions of partial Rx-OTC switch timing is evidently

needed but, so far, not found in the extant operations management literature. This paper aims to fill this

gap. Specifically, we formulate and solve a model of the proactive partial Rx to OTC switch timing

optimization problem by a profit-maximizing firm. The solution establishes when it would be optimal for

a monopoly (patent-protected) brand to make the partial switch and, more specifically, its precise time

relative to the Rx patent expiration date. Further, the proposed model permits us to investigate the

following outstanding questions: (1) Is the grant of 3-year market exclusivity necessary to induce the firm

to make a partial switch before the Rx drug’s patent expiration date? (2) What is the optimal partial

switch time with and without 3-year market exclusivity? (3) What are the corresponding implications for

the firm’s profits and consumers’ (payers’) savings in Rx drug expenditures?

The rest of the paper is organized as follows. In §2, we provide a brief review of related health

economics research and existing new product launch timing models, and how our model contributes to

these research streams. In §3 we formulate the partial Rx-OTC switch timing problem incorporating

general Rx and OTC user growth dynamics as an optimal control problem. In §4, we derive the solution

to the general problem formulation and show that, counter to conventional thinking, the grant of 3-year

market exclusivity is neither necessary nor sufficient for it to be optimal to partially switch before the Rx

drug’s patent expiry. To investigate our other research questions, in §5 we solve a functionally specified

case of our general model and numerically examine the optimal partial switch timing with and without 3-

year market exclusivity, utilizing realistic sample model parameter values drawn from historical cases.

In §6, we discuss the implications of our findings for partial switch planners and regulatory policymakers

and conclude with a summary of the paper’s contributions and directions for future research.

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2. Previous Research

Related Economics and Health Economics Research Literature

The health economics literature on Rx to OTC switch strategies is relatively sparse. Berndt et al.

(2000) have done empirical research on ‘sunset’ antiulcer brands, namely, Pepcid, Zantac, Tagamet, and

Axid, that made such switches over the 1989-1998 decade period. They found substantial interactions

between the Rx and OTC market segments, with direct-to-consumer marketing efforts playing a major

role, and brand OTC margins were 30-50% of their brand Rx margins (see also Lichtenberg 2003). Other

empirical health economics papers have focused on Rx to OTC switch impact on consumer welfare and

managed care drug utilization (e.g., Temin 1992, Gurwitz et al. 1995, Harris et al. 2005). These papers

typically find that Rx to OTC drug switches can be beneficial to consumers and help to lower prescription

drug bills. However, as noted by Shih et al. (2002) this literature has concentrated on complete rather than

partial switches. In particular, the impact of 3-year OTC market exclusivity on partially switched Rx drug

spending has not been explored as we do in this paper.

Marketing Timing Models Literature

Several management science papers, e.g., Kalish and Lilien (1986), Lilien and Yoon (1990),

have investigated the optimal timing of a new product introduction, but only a few have considered the

introductions of extensions of existing products into existing or new markets or channels. Notably,

Wilson and Norton (1989) and Mahajan and Muller (1996) examine the optimal timing of introduction of

successive generations of a high technology product into the same market while papers by Lehmann and

Weinberg (2000), Prasad, Bronnenberg, Mahajan (2004) investigate the optimal timing of entry of

essentially the same product (a movie) into successive distribution channels. The Rx to OTC partial

switch problem treated by us, however, involves elements of both. Also, the previous papers have

assumed very long or infinite planning horizons. In contrast, our focus on is on a finite horizon

optimization problem with definite end points. In general, solving such problems is significantly different

and more complicated than solving an infinite horizon problem, and typically involves numerical analysis

(e.g., Bass, Krishnamoorthy, Prasad and Sethi 2005, Raman 2006). Additional novel, albeit complicating,

features of our problem are that it involves two phases, the single Rx drug phase followed by the dual

phase with interacting Rx and OTC dynamics, and a phase switching cost as well as salvage values at the

Rx patent expiration date that are time-and state-dependent. By formulating and solving this new finite

horizon, two-phase dynamic optimization problem (cf., Amit 1986, also, Kamien and Schwartz 1991), our

research contributes to both the marketing timing models and health economics literatures.

3. Model Formulation, Notation and Assumptions

We consider a pharmaceutical company which has launched a Rx drug brand at time t= 0 and

whose patent expires at t = T. We suppose that the drug belongs to a class of drugs, e.g., anti-obesity,

which treat a recurring indication such as weight gain that is amenable to self-medication by consumers.

We further suppose that the drug is the first and only Rx version of its chemical entity type approved for

marketing in this class (e.g., Xenical). We take it that the firm’s brand management has identified a

specific therapeutic opportunity for launching an OTC version of the Rx drug, e.g., the same or milder

form of a current indication, and the question now is whether and when to make a partial switch before T.

Dynamics of Users: A General Formulation

Let x=x(t) and y=y(t) be the cumulative number of users of version Rx and OTC, respectively, at

time t. By ‘users’ we are referring to the consumers of the drug who have determined that the drug works

satisfactorily for them (e.g., via a sample prescription from the physician in the case of the Rx version or

by a self-medicated trial in the case of the OTC version). Users continue to take the drug for that ailment

whenever it recurs until they exit the market, because of death, finding a permanent cure, emigration etc.

Let anddx dyx ydt dt

= = denote the instantaneous rate of change in the cumulative number of

users of version Rx and OTC, respectively. Let τ be the partial switching time. The periods [0,τ] and [τ,T]

are the single status (Rx drug only) and dual status phases, respectively. We assume that the rate of

change in the number of Rx drug users at any instant in the single status, is given by

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x = ( )f x x(0)=0, 0 t .τ≤ ≤ (1)

Upon integration, Equation (1) can be rewritten in the form,

x=F(t,τ ), 0 t .τ≤ ≤ (1a)

In the dual status phase, we assume that the rates of change in the number of Rx and OTC drug

users at any instant are given by

( , ), ( ) 0, .

( , ), ( ) ( , )t

y g x y yt T

x h x y x F tτ

ττ

τ τ=

= = ⎫≤ ≤⎬= = ⎭

(2)

Note that the general formulations in Equations (1) and (2) allow for a variety of dynamic specifications

that can incorporate not only growth but also decay in the numbers of users as shown in the functionally

specified illustration in Section 5. Since for our purposes the functions g and h, and their partial

derivatives with respect to x and y, are continuous there is a unique solution that satisfies the initial

conditions (see Boyce and Diprima 1986). Let these solutions be denoted

x=G(t,τ ) and y=H(t,τ ), .t Tτ ≤ ≤ (2a)

Firm’s Profits

As the drugs treat (or prevent) ailments that are recurring, the firm obtains ongoing revenues from

each satisfied user of its products as s/he periodically consumes the prescribed dosage of the drug. For

example, consumers take one 10 mg tablet of Claritin or Zyrtec every day to forestall allergic attacks or

one tablet of Pepcid before every meal to prevent heartburn. Let and denote the quantities

consumed per unit time on average by Rx and OTC version users, respectively. Typically, when Rx and

OTC forms of the drug are simultaneously available, > as the Rx user is likely to be more

seriously afflicted. Given these consumption rates, we next formulate the firm’s cumulative discounted

profits over the single status and dual status phases, accounting for marketing expenditures. Let

xD yD

xD yD

1Π be

the firm’s cumulative discounted profit from the single status phase. Then,

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1 1 10

( ) rtxm x C e dt

τ−Π = −∫ , (3a)

where r denotes the discount rate, =q1xm 1xDx, and q1x and C1 respectively denote gross profit rate from

drug Rx and the marketing spending in the single status. Note that the profit function implies an ongoing

stream of revenues (as a result of repeated use) from each user. Next, let 2Π be the firm’s cumulative

discounted profit from the dual status phase. Then,

2 2 2(T

rtx ym x m y C e dt

τ

−Π = + −∫ ) , (3b)

where m2x=q2xDx , my=qyDy and , q2xq y and C2 , respectively denote gross profit rate from drug Rx, gross

profit rate from drug OTC, and the marketing spending in the dual status.

We now consider the firm’s single to dual status switching cost, denoted SC. One component of

this phase-switching cost is the firm’s cost of conducting in-use trials and studies that provide sufficient

evidence to satisfy the FDA that the product is effective and safe for nonprescription use (see, e.g., The

Food & Drug Letter, #666, 2002). We expect this cost to decline as the number of Rx users increases.

That is, as more evidence of the product’s efficacy and safety accumulates, fewer and/or less intensive

additional tests with the OTC version would be needed to gain FDA approval for its marketing (The Food

& Drug Letter, #666, 2002). However, a second component of the switching cost is the cost associated

with aging of the Rx brand’s therapeutic technology and/or value proposition which makes it more

difficult to position the drug as a “new” treatment in the OTC market. Based on the above considerations,

therefore, we express the overall partial switch time-dependent phase switching cost value as,

SC= 0

( )l

x τ 1lτ+ . (3c)

where l0 > 0 and l1 > 0 are respectively the equity-related and age-related switching cost constants.

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Next, extant empirical research studies, e.g., Cook (1998), have shown that the Rx drug market

has some salvage value ($) after the Rx patent expires. Let S(x (T)) denote this salvage value, expressed

as:

(3d) ( ( )) ( ).xS x T s x T=

In (3d) , sx , represents an assessment of the salvage value ($) per Rx user (in relation to some specified

salvage horizon, e.g., six months or one year) at T.

Similarly, let be the firm’s post-T salvage value ($) from the OTC market. The

revenues from the OTC version after a partial switch also do not vanish at T because of brand-loyal, self-

medicating, and self-paying users in that market. Then,

( ( ))S y T

( ( ))S y T = ( )ys y T . (3e)

In (3e), y(T) is number of OTC users active at T, and denotes the assessment of the salvage value ($)

per OTC user at T, and is given by,

ys

ys = 0δ + 1[1 sgn( )]( )

2e

et Tt T τδ τ + + −

+ − , 0,0 10 >> δδ (3f)

where te denotes the OTC market exclusivity period length. (In (3f), sgn is 1 for a positive argument and -

1 for a non-positive argument). Thus, we assume the salvage value per OTC version user increases with

the length of the OTC market exclusivity period remaining at T, i.e., et Tτ + − . More specifically, the

parameter δ0 can be interpreted to be the brand’s head start value per OTC user active at T because, until

the Rx drug patent expires, a generic or branded competitor’s application for marketing a copycat OTC

version cannot be processed by the FDA. Moreover, the FDA’s processing itself takes some additional

time before this application is approved (see, e.g., Glover 2007). In the meantime, the branded OTC

version enjoys the advantage of its head start within the Rx patent-protected period. We assume the

magnitude of δ0 becomes larger as the expected time for filing of the competitor drug’s application and/or

its processing by the FDA increases. Additionally, if the branded firm obtains 3-year OTC market

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exclusivity, the FDA cannot begin to process a copycat OTC competitor’s application until this

exclusivity period expires. Then, δ1 represents the associated exclusivity-added value of an OTC user per

year of market exclusivity remaining at T. As we show later, in determining when to switch, a key

tradeoff that planners must make is that between the head start and post-T exclusivity-added advantages.

Let Π be total discounted profits of the firm over the finite horizon [0,T]. Combining (3a)-(3e),

we obtain

01 1 1 2 2

0

( ) ( ) ( ) ( ) .( )

Trt r rt rT

x x y ylm x C e dt l e m x m y C e dt s y T s x T e

x

ττ

τ

ττ

− − −⎡ ⎤⎡Π= − − + + + − + +⎢ ⎥ ⎣

⎣ ⎦∫ ∫ x

−⎤⎦ (3g)

4. Firm’s Partial Switch-Timing Decision Problem

The firm needs to choose τ to maximize Π, defined in (3g), given the dynamics in (1) and (2), or

01 1 1 2 2

0

0 1

max

( ) ( ) ( ) ( ) ,( )

[1 sgn( )] ( ) ,2( , ), 0

( , ) ( , ), .

Trt r rt rT

x x y y

ey e

lm x C e dt l e m x m y C e dt s y T s x T ex

t Twhere s t T and

x F t tx G t and y H t t T

τ

ττ

τ

ττ

τδ δ τ

τ ττ τ τ

− − −

Π ⎫⎪⎪⎡ ⎤

⎡ ⎤= − − + + + − + + ⎪⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎪

⎪+ + −= + + − ⎪

⎪⎪= ≤ ≤ ⎬⎪= = ≤ ≤ ⎪⎪⎪⎪⎪⎪⎪⎭

∫ ∫ x−

(4)

To solve this two-phase optimization problem we use a variational approach. Next, we formulate the first-

order necessary conditions for the solution.

Necessary Conditions

Let )(1 tλ and )(2 tλ be the current multipliers of x in periods 0 t τ≤ ≤ and t Tτ ≤ ≤ ,

respectively, and )(1 tµ be the current multiplier of y in the period t Tτ ≤ ≤ . We assume these

multipliers are continuous functions. In our context, the current multipliers represent shadow prices

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associated with a unit change in the total number of users of drug i, i.e., the net benefit (or loss) to the

firm of having one additional user buying the drug at time t. Then, we obtain the following result.

THEOREM 1. Consider the problem of profit maximization over the Rx brand planning horizon [0,T] encompassing two phases described by (4). Then the optimal launch time of OTC version should satisfy the following conditions: (1) There exist )(1 tλ , )(2 tλ and )(1 tµ which satisfy

1 1rλ λ= - -1xm 1λfx∂∂

, (5)

2 2rλ λ= - -2xm 2λhx∂∂

- 1µgx∂∂

, 2 ( ) xT sλ = (6)

1 1rµ µ= y- m - 2λ hy∂∂

- 1µ gy∂∂

, 1( ) yT sµ = =0 1

[1 sgn( )]( )2

ee

t Tt T τδ δ τ + + −+ + −

(7)

and 01 22 ( ) ( )

( )l

xλ τ λ τ

τ− + =0. (8)

(2) Let

L(τ )= C2-C1+ 022( ( )) ( )f x + 0

1 1( )( )lr l

xτ

τl+ −

( )l

xλ τ

τ+

2 ( )λ τ− ( ,0)h x 1( ) ( ,0)g xµ τ− ( )1

sgn( ) 1( )2

r T et Ty T e τ τδ − − + − ++ (9)

At the launch time,τ , of the OTC version, the following condition must hold

( )Lτ =0. (10)

Considering (2b), (6) and (7) form a system of first-order linear differential equation and since the

functions in RHS of (6) and (7) and their partial derivatives with respect to 2 , 1λ µ are continuous

functions, they have a unique solution for the period t Tτ ≤ ≤ (see Boyce and Diprima 1986).

Substituting the values of 2 ( )λ τ and x(τ ) from (1b) into (8) leads to 1( )λ τ that is the boundary

condition of Equation (5). Similarly considering (1b), Equation (5) together with its boundary condition is

a first-order linear differential equation that has a unique solution in [τ,T]. Substituting the values 2 ( )λ τ ,

1( )µ τ and x(τ ) from (1b) into (9) leads to a condition for the optimal timing τ (See illustration in § 5) .

Condition (10) bears on our focal optimal partial switch timing questions. Let

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1 1 1 1( ) ( ) ( ) ( ( ))xH m x C f xτ τ λ τ= − + τ , and H2 ( )τ = 22 ( ) ( )x ym x m y Cτ τ+ − + 2( ) ( ( ))h xλ τ τ + 1( ) ( ( ), ( )).g x yµ τ τ τ

In this notation (10) becomes

( )01 1 1 2 1

sgn( ) 1( ) ( ) ( ) ( )( ) 2

r T el tH r l l H y T ex

τ Tττ τ τ δτ

− − + − ++ + − = −

.

Thus (10) means that if there is a time τ at which H1 ( )τ less the marginal cost of switching

01[ ( ) ]

( )lr l

xτ

τ− + − 1l from the single to dual status equals H2 ( )τ less the marginal terminal value then it

is optimal to switch at τ . If such a switch occurs then according to (8), at this point of time, the valuation

of x in the single status, 1λ , plus the marginal switching cost with respect to x must equal the valuation of

x in the dual status, 2λ . Also, as indicated by conditions (6) and (7), the marginal valuation of x at T is

naturally , while the marginal valuation of y at T is, Txq

ys = 0 1[1 sgn( )]( )

2e

et Tt T τδ δ τ + + −

+ + − . (11)

Analysis

We now focus on Condition (10) for the optimal partial switch time.

REMARK 1: In (10) since L(τ ) is continuous, it follows that the necessary and sufficient conditions that τ will be an optimal switch time is that

L(τ )=0 but L(τ -) > 0 and L(τ +) < 0.

To investigate if and when it would be optimal to launch the OTC version before the patent

expiration date T, we first examine the implications of L(τ ) at τ =0+ and τ =T. Considering the initial

condition in (1), it is easy to see that,

L(0+) > 0. (12)

Next, at t=T, we have

L(T)=C2-C1+ 02 ( ( ))( )l f x T

x T+ 0

1 1[ ]( )lr l T l

x T+ − xs- ( - )-( ( ),0)h x T ( ( ))f x T 0 1( )etδ δ+ ( ( ),0)g x T . (13)

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Using these findings we arrive at the following propositions that bear on our research questions.

PROPOSITION 1. Consider the problem of profit maximization over the Rx brand planning horizon [0,T] encompassing two phases described by (4). Then, it is never optimal to switch to a dual strategy as soon as Rx is introduced even if the firm has arranged the approval for an OTC version at this time.

PROOF: See Appendix.

PROPOSITION 2. Consider the problem of profit maximization over the Rx brand planning horizon [0,T] encompassing two phases described by (4). If the condition

L(T) < 0 (14) holds, then, it is optimal for the firm to switch to the dual strategy before the Rx patent expires.

PROOF: See Appendix.

PROPOSITION 3. Consider the problem of profit maximization over the Rx brand planning horizon [0,T] encompassing two phases described by (4). If condition (12) is not satisfied at any point (0, )Tτ ∈ then the firm should stick entirely with the Rx drug and not switch to the dual strategy until the Rx patent expires, if the following condition holds:

L(T) (15) 0≥PROOF: See Appendix.

Thus, Proposition 2 gives us the answer to the “Should the OTC product ever be launched before

the Rx patent expiry?” question, while Proposition 3 provides the condition for the firm not to make the

Rx to OTC switch before T.

Impact of Additional 3-Year Market Exclusivity on “Whether or Not to Switch before T?”

Using (13), let us rewrite the condition (14), as follows:

0( ) 0

1 1 2 1( ) ( )0 1

( ( )) [ ( ( )) ( ( ),0)] ( )(16)

( ( ),0)

lx T l

xx T x Te

f x T s f x T h x T r lT l C Ct

g x Tδ δ

+ − + + − + −+ >

Upon examining (16), we see that the condition for making the switch at T is that the incremental gain in

salvage value from the OTC market should exceed the incremental total cost of making the switch at T,

the latter being the sum of the phase-switching cost at T, the reduction in this switching cost forgone by

not waiting to acquire an additional Rx user, the decrement in the salvage value from the Rx market by

introducing the OTC drug at T, and the incremental marketing cost.

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Hereafter we shall refer to the RHS of (16) simply as the partial switch cost hurdle. Denoting

this hurdle value by F, i.e.,

F=

0( ) 0

1 1 2( ) ( )( ( )) ( ( ( )) ( ( ),0) ( )( ( ),0)

lx T l

xx T x T 1f x T s f x T h x T r l T l C Cg x T

+ − + + − + −,

the following corollaries summarize this analysis:

COROLLARY 1. If δ0+δ1te, the salvage value per OTC user when the OTC drug is launched at the Rx expiration date, is greater than the hurdle F then a pre-patent expiry OTC launch will be optimal. However, if 0 1 etδ δ+ ≤ F then the firm should never do a pre-expiry OTC launch. COROLLARY 2. Suppose the value of the hurdle F is fixed. Then a higher value of 0 1 etδ δ+ can induce a firm to change its optimal timing decision from “Never before T” to “At or before T” and conversely, a lower value of 0 1 etδ δ+ can induce a firm to change its optimal timing decision from “At or Before T” to “Never before T.” Considering Corollary 2, for a fixed value of the hurdle F, the case for a partial switch before T is

strengthened (weakened) by:

(i) A provision (no provision) of 3-year OTC market exclusivity. (ii) A higher (lower) OTC head start advantage (δ0) per OTC user. (iii) A higher (lower) incremental value (δ1) per year of remaining market exclusivity per OTC

user.

Notably, if the head start advantage δ0 is sufficiently large, it can be optimal for the firm to make the

partial Rx to OTC switch before T even if market exclusivity for the OTC drug is denied.

COROLLARY 3. Suppose the value of 0 1 etδ δ+ is fixed. A lower value of the hurdle F can induce a firm to change its optimal timing decision from “Never before T” to “At or Before T” and conversely, a higher value of F can induce a firm to change its optimal timing decision from “At or Before T” to “Never before T.” Considering Corollary 3, for a fixed value of 0 1 etδ δ+ the case for a partial switch before T is

strengthened (weakened) by:

(1) A lower (higher) single status Rx user acquisition rate, , at T. f(2) A higher (lower) dual status Rx user acquisition rate, h , at T. (3) A higher (lower) number of accumulated Rx drug users, x(T), at T. (4) A lower (higher) salvage value per Rx user in the Rx market (assuming > , at T). T

xq f h

15

(5) A higher (lower) OTC user acquisition rate, g , at T. (6) A lower (higher) value of the phase switching cost proportionality constant,l0. (7) A higher (lower) value of the aging-related switching cost proportionality constant l1. (8) A lower (higher) added cost of marketing the OTC product at T. (9) A lower (higher) value of the discount rate r. 5. Analysis of a Specified Form of the Partial Rx-OTC Switch-Timing Model

In this section, we analyze a functionally specified form of the general model presented in the

previous sections. Within the context of this model, our objectives are three-fold: (1) explicitly answer

the “Exactly When” to make the partial switch question; (2) demonstrate how the provision or denial of

three-year OTC market exclusivity impacts the optimal partial switch-timing using a numerically

parameterized specification of the model; and (3) examine how 3-year OTC market exclusivity impacts

the firm’s profits and consumers’ (payers’) welfare, specifically savings in Rx drug expenditures.

Specification of User Evolution Equations

Let denote the potential market size or number of Rx drug users in the populace in the single

status Rx phase, and N

xN

y denote the total number of potential users of either version of the brand molecule

in the dual status phase. We assume Ny ≥ Nx, (as the OTC version launch increases the size of the

potential market by drawing in consumers solely interested in self-medication). Then, we specify the

continuous-time user growth equations in the single and dual status phases as follows:

1( )x x xx a N x xδ= − − , x (0)=0, 0 t τ≤ ≤ (17a)

1

1 2

( )1 2 2

1

(1 )[ ( )] , ( ) 0

[ ( )] , ( ) (1 )x x

y y y y

ax xx y y x

x x

y a N x y a x y yt Ta Nx a N x y a x x x e

aδ τ

α α δ ττ

α α δ τδ

− +

⎫= − − + + − =⎪ ≤ ≤⎬= − + − − = − ⎪+ ⎭

(17b)

Equation (17a) implies the instantaneous change in Rx users in the single status phase is given by

new users drawn from the untapped pool xN x− at the rate ax, 0 < ax < 1, less dropouts per instant from

among current users at the rate 1xδ , 0 < 1xδ < 1 (due to reasons such as death, migration, lifestyle change

etc).

16

Next, in the dual status phase, Equations (17b) imply that a fraction 0 < 1α < 1 of the remaining

number of potential users of the medication, (yN x y)− + , will be interested in only the Rx version of the

drug, perhaps because they are more seriously afflicted consumers who require the prescription strength

medication and/or suffer from the indications reserved for the Rx drug as in the case of Lamisil topical

ointment. Therefore, (1-α1) [ ] denotes the untapped potential market for the OTC version.

At the same time, Equations (17b) assume a fraction α

(yN x y− + )

2 of the current Rx drug users x discontinue

treatment with the Rx drug and switch over to the OTC product. This could be because the latter is

sufficient for their indications or because their health insurance plan stops or restricts reimbursement of

the Rx drug due to the availability of an OTC version as occurred in the case of Claritin (Trygstad et al.

2006). Thus, the parameters (1-α1) and α2 represent cannibalization of existing and potential Rx drug sales

by the new OTC product. Additionally, Equations (17b) also allow for continuous decay in the numbers

of current users of Rx (at the rate 2xδ ) and OTC users (at the rate yδ ) in the dual phase. Lastly, in

Equations (17b), ay, reflects the direct positive influence of OTC advertising and the spillover effect of Rx

brand reputation on generating OTC users (e.g., Berndt et al 2003); xa reflects the direct positive

influence of Rx drug marketing efforts, e.g., detailing and DTC advertising, on creating Rx drug users;

and ya reflects the effect of OTC product advertising in inducing existing Rx users to switch to the OTC

product (e.g., Datamonitor 2000).

Partial Switch Timing Solution

Substituting Equations (17a) and (17b) in (4), the general model of the firm’s optimal partial

switch timing problem, Equations (5)-(8) of Theorem 1 become,

1 1rλ λ− = - +1xm 1λ 1( x xa )δ+ , (18a)

2 2rλ λ− = - +2xm 2λ 1 2 2( )x y xa aα α δ+ + 1µ− 2 1( (1 )y ya a )α α− − , 2 ( ) xT sλ = (18b)

=− 11 µµ r 2 1 1 1[(1 ) ]y x y y

17

m a aλ α µ α δ− + + − +1 0 1

[1 sgn( )]( ) ( )2

ee

t TT t T τµ δ δ τ + + −= + + − (18c) ,

and, 01 22 ( ) ( )

( )l

xλ τ λ τ

τ− + = 0 . (18d)

Equations (18b), (18c) form a system of first-order linear differential equations that can be integrated to

obtain λ2(t) and µ1(t) for the period .t Tτ ≤ ≤ Substituting the values λ2(τ ), µ1 (τ ), x(τ ) and the

specifications (17a)-(17b) into (9) will lead to a condition for the optimal timing τ that depends on the

parameters of the model.

Insights into the “When before T Partial Switch” Question.

The following proposition explicitly answers the “Exactly When” to make the partial switch

question.

PROPOSITION 4. Consider the problem of profit maximization over the Rx brand planning horizon [0,T] encompassing two phases described by (17a-b), and let,

L(τ )= +2 1`( )C C− 022[ ( )] 1[ ( ( )) ( )]x x xa N x xτ δ τ− − 0

1 1( )( )lr l

xτ

τl+ + −

( )l

xλ τ

τ+

2λ− (τ ) 1 2[ ( )] ( ) (x y y xa N x a x2 )α τ α δ τ− − +

1µ− (τ ) 1 2(1 )[ ( )] ( )y y ya N x a xα τ α τ⎡ ⎤− − +⎣ ⎦ ( )1

sgn( ) 1( )2

r T et Ty T e τ τδ − − + − ++ ,

(19) where 2 ( )λ τ and 1µ τ ) are the solutions of (18b)-(18c) at t=τ . Then the firm should switch to a dual strategy at τ if and only if

L(τ ) = 0, L(τ -) > 0 and L((τ +) < 0. (20)

PROOF: See Appendix.

Applying Proposition 4 to Investigate the Impact of 3-year Market Exclusivity on τ

In this illustration, we numerically investigate the influence of the 3-year market exclusivity

provision on τ under different market scenarios. Each scenario is defined by some combination of fixed

ratios of OTC:Rx prices and market sizes, two key dimensions of any switch opportunity (e.g., Mahecha

2006). More specifically, we assume the Rx drug’s full price, hereafter denoted px, remains the same

whether or not an OTC version is launched, i.e., p1x = p2x = px and m1x = m2x = mx.. We arbitrarily set px =

18

$1.05 per unit for a payer (whether that be an insurer or an uninsured consumer), and assume the

production cost per unit, whether it be sold in the Rx or OTC markets, is constant and equal to $0.25. This

implies the Rx drug’s gross margin is about 76% which is quite typical for Rx drugs under patent (e.g.,

King et al. 2000). To set the range for the OTC version’s price, denoted py, we use, as is commonly done

(see, e.g., Spencer 2000), the average copayment for the Rx version made by consumers with and without

private insurance as the lower and upper bounds. For example, a study by Lichtenberg (2003) reports this

range to be 28.5% to 75% of the full price in the case of the antihistamine Rx drug Zyrtec. Based on these

observations, we shall assume alternative OTC price levels of $0.3, $0.35, $0.43, $0.5, and $0.75, i.e.,

OTC:Rx price (margin) ratios of 0.285 (.0625), 0.333 (0.125), 0.41 (0.225), 0.476 (0.3125), and 0.714

(0.625) in our illustrative analyses. (Hereafter, we use the notation pyx = py/ px and Myx = my/mx to refer to

these ratios.)

Next, prior surveys (e.g., a survey of consumers’ self-medication behavior by Roper Starch

Research 2001) and case studies (e.g., King et al. 2000) suggest that for the most common ailments like

headaches, heartburn/indigestion, allergies, and skin disorders, potential market sizes for the OTC version

of Rx drugs can range from two to five times the size of the Rx drug’s potential market. Therefore, we

vary the Nyx = / ratios in our numerical illustration between 2 to 5. Specifically, we arbitrarily set

= 1100, and consider levels of 2200, 3300, 4400, or 5500 in our scenarios. Lastly, the empirical

sources and justification for the selected values of all other model parameters in this investigation are

detailed in Appendix Table A1. In particular, based on Grabowski’s (2002) findings, we take the

effective total length of the Rx patent horizon is T = 12 years.

yN xN

xN yN

Optimal partial switch times results

Applying Proposition 4 to our numerically specified model, we can derive the firm’s optimal

switch times τ with and without 3-year market exclusivity, hereafter denoted τ3 and τ0 respectively, for

different selected combinations of Myx and Nyx ratios. Figures 1(a)-1(d) display the results graphically. As

19

20

indicated in Figure 1(a), when Nyx is as small as 2, and te = 0, it is not optimal to switch before T=12 at

any of the selected levels of Myx, illustrating our earlier Proposition 3. However, when te = 3, we see that

τ3 < 12 even when the Myx value is as low as 0.0625. Further, the optimal switch timing advances as the

Myx ratio increases in value. Next, Figure 1(b) shows that when Nyx = 3, both τ0 and τ3 occur before T

when Myx exceeds 0.25. Further, at At higher Myx values, τ0 and τ3 begin to converge and are equal when

Myx is as high as 0.65. However, when the Nyx ratio is increased to 4 (Figure 1(c)), we observe that τ0 and

τ3 are < 12 at all Myx levels, and advance as these increase, but τ3 < τ0 for the low Myx values of 0.0625 and

0.125, while τ0 < τ3 at the higher Myx value of 0.225. That is, there is a reversal in the order of τ0 and τ3

over the plausible range of Myx. Again, τ0 and τ3 converge to the same value at Myx = 0.25. Lastly, as

shown in Figure 1(d), τ0 and τ3 are even more advanced in time at all Myx when Nyx is as high as 5. Again

there is reversal in their order when Myx is in the region of 0.125, and convergence when Myx becomes

0.25. Notably, at the very plausible Myx level of 0. 225, τ0 is as early as 6.6 years before patent expiry

(similar to the timing of introduction of Lamisil’s OTC version by Novartis.)

These results offer several interesting and important insights for pharma switch planners. First

and foremost, the common heuristic that, given 3-year OTC market exclusivity, it is best to make the

partial switch a short time (1 to 2 years) before the brand name Rx drug patent expires is clearly too

simplistic. Second, our results run counter to the intuitive view of many industry executives that it is not

optimal to switch to dual status without 3-year OTC market exclusivity (see, e.g., Pfister 2004,

Harrington and Shepherd 2002). As shown in Figure 1, it can be optimal to launch the OTC version

before T even if the 3-year OTC market exclusivity is denied (see Corollary 2 in general analysis).

Indeed, in some of the intermediate scenarios, it is optimal for the firm to launch the partial switch even

sooner without market exclusivity than with it. Intuitively, the firm achieves greater capitalization of the

head start advantage by advancing the partial switch time in order to accumulate as large a number of

OTC users by T as is financially worthwhile. On the other hand, with 3-year OTC market exclusivity,

the firm is induced to launch the OTC version later in order to extend the OTC exclusivity-added

advantage δ1(τ +te–T) after T and also increase its salvage value from the Rx drug market as a result of

staying longer in the Rx only single status phase.

Impact of 3-year market exclusivity provision on Firm’s and Rx Drug Consumers’ Welfare

Our analysis in this section is restricted to the patent-protected horizon of length T. In general,

FDA approval for a partial switch is clearly an opportunity for the firm to make increased profit from the

expanded Rx plus OTC market. However, as Figure 1 indicates, if the potential OTC market size is

relatively small, e.g., Nyx = 2, it is more profitable for the firm not to make a partial switch before T ,

unless OTC marketing approval is accompanied by the provision of 3-year OTC market exclusivity.

However, the need for this incentive apparently diminishes as Nyx increases, and Myx becomes more

attractive. Further, as Figures 1(c) and 1(d) show, when the expanded Rx+OTC market size becomes

sufficiently large, it is optimal for the firm to switch at such an early point in the patent-protected horizon

that the 3-year market exclusivity becomes redundant. Of course, at intermediate levels of Nyx and Myx,

the firm always makes higher profit when te = 3. To illustrate, Table A2 shows the firm’s numbers of

users and profit outcomes, with te = 0 or 3, for the Nyx = 4 case, compared to the benchmark ‘No Switch

before T’ case. We see that the firm’s total profit gain is higher when te = 3 and increases as Myx is

increased. But this difference disappears when Myx exceeds .3125 and the firm switches early regardless

of te.

But does the provision of 3-year OTC market exclusivity to the firm also benefit Rx drug users or

payers by way of savings in Rx drug expenditures? To shed light on this, we compare the total spending

over T by Rx drug only users in the ‘no switch before T’ case with the corresponding total Rx drug

expenditures with and without the grant of 3-year market exclusivity to the firm (see formulae in

Appendix). The results in the case of Nyx = 4 are shown in the last column of Table A2. We see that

allowing the firm to make a partial switch as such does yield positive savings to Rx drug consumers.

However, the need for 3-year market exclusivity to accompany OTC marketing approval is questionable.

Consider, e.g., the difference in Rx consumer savings when te =3 versus te = 0 for the case when Nxy = 4

21

22

(Figure 2). Specifically, we see that while te=3 induces the firm to partially switch and thereby generate

some higher savings for Rx drug consumers at very low pyx values, this provision lowers their savings,

i.e., makes them worse off, at intermediate and plausible OTC price levels because it induces the firm to

launch the OTC version later than it would without it.. To summarize, it is only when OTC:Rx market

sizes and/or margin ratios are rather low that the provision of 3-year OTC market exclusivity improves

both the firm’s and the Rx drug consumers’ lots compared to the case when this incentive is denied. In

other cases, this provision works only to the advantage of the firm and not Rx drug consumers.

6. Managerial Implications and Conclusions

The general and functionally specified analyses of our model have the following implications.

First, they rigorously establish the conditions under which partial switches before Rx patent expiry can be

optimal, thereby contributing to a more informed debate between advocates and opponents of this strategy

within firms. In particular, our analysis shows that the OTC market exclusivity-added benefit should not

overshadow the inherent OTC marketing head start advantage accruing from launching the OTC version

within the Rx drug’s patent-protected period. The head start advantage alone may be sufficient for a

partial switch before T to be optimal. Therefore, from a proactive planning viewpoint, branded pharma

managers should concentrate on assessing the OTC market opportunity from an early point in the Rx

brand’s patent-protected lifecycle and, if deemed worthwhile, be prepared to move forward whether or

not the FDA grants 3-year OTC market exclusivity.

Operationally, our research offers an implementable optimization model framework that can be

readily used by firms’ planners to make such assessments provided historical and primary market

research-based estimates of all relevant parameters. The functionally specified model can be used

investigate a vast array of “what-if” scenarios to better understand how optimal switch timing varies with

changes in various parameter values.

Figure 1: Optimal Timing of Proactive Partial Switch

F

igure 2: Difference in Savings in Rx Drug Spending with te = 3 vs. te = 0

23

From the viewpoint of benefit to Rx drug consumers, however, our analyses raise questions about

the Waxman-Hatch Act’s provision of 3-year market exclusivity to encourage firms to make Rx to OTC

drug switches as a way to lower payer and consumer medication costs. We have shown that when the

OTC market size is relatively attractive, this provision is a “nice to have” but not a “must have”

inducement for firms to proactively plan a partial switch. In such scenarios, the savings in consumer Rx

drug spending are actually greater when the firm is denied 3-year OTC market exclusivity as it removes

the firm’s incentive to delay the launch to a point closer to the patent expiration date. OTC market

exclusivity provision is most beneficial to the firm as well as Rx drug consumers when the market

opportunity is relatively small rather than large.

In conclusion, this paper contributes to the general optimal marketing timing models literature

and to the growing literature on prescription drug lifecycle management strategies. From the viewpoint of

future research, the model assumes the Rx drug molecule is the first in its class facing no competition

from similar Rx drug brands. Extending the model to categories where there are multiple Rx drug makers

deciding if and when to introduce OTC versions of their existing Rx drugs would be an important and

worthwhile direction for future research. We hope this work serves as a catalyst for such studies.

Appendix

Proof of Theorem 1:

Adjoining the constraint equations in (4), to Π with the Langrange multipliers, 121 ,, µλλ , we obtain,

1

1

1

01 1 1 2 2 1

0

[ ( )] ( ) [ ( ( )] ( ( ) ( ))( )

t Trtrt rt rT

y xt

lL f x e dt l e L h x g y e dt s y T s x T ex

λ τ λ µτ

−− −Π= + − − + + + − + − + +∫ ∫ − , (A1)

where L1= , 1 1xm x C− 0 t τ≤ ≤ and L2= 2 2x ym x m y C+ − , t Tτ ≤ ≤ . For convenience we define the scalar

functions H1 and H2 (the current Hamiltonians), as follows,

H1= +1 1xm x C− 1λ ( )f x , 0 t τ≤ ≤

24

25

x ym x m y C H2= 2 2 2λ+ − + ( , )hx y + 1µ ( , )gxy , t Tτ ≤ ≤ .

Then, integrating the remaining terms in (A1) by parts, yields

1

1

01 1 1 1 1 1

0

2 2 2 1 1

2 2 1 1

[ ( ) )] ( ) ( ) ( ) (0) (0)( )

[ ( ) ( ) ]

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( )) .

trt r r

Trt

t

rT r rT r rTy x

lH r x e dt l e e x xx

H r x r y e dt

T e x T e x T e y T e y s y T s x T e

τ τ

τ τ

λ λ τ λ τ τ λτ

λ λ µ µ

λ λ τ τ µ µ τ τ

− − −

−

− − − −

Π = + − − + − +

+ + − + −

− + − + + +

∫

∫−

Now consider the first-order variations in Π due to variations in the control parameter τ, and the state

variables x and y, for fixed initial conditions, x(0)=0 and y(τ)=0, and fixed initial time t=0, and terminal

time, T,

1

1

( )01 1 1 2 1 1 1 1

0

02 1 1 2 2 2 2 1 12

sgn( ) 1( ) ( ) [( / ) ]( ) 2

( ) ( ) (0) (0) [( / ) ( / ) ]( )

(

r T t rt rtet

Trtrt rt rt

tt

l t t TH r l t l H y T e e d H x r x e dtx t

l e e e x t x H x r x H y r y e dtx t

q

τ

τ

τ

δ δ τ λ λ δ

λ λ δ λ δ λ λ δ µ µ δ

− − − −

=

−− − −

=

⎛ ⎞+ − +Π= + + − − + + ∂ ∂ + −⎜ ⎟

⎝ ⎠

+ + − + + ∂ ∂ + − + ∂ ∂ + −

+

∫

∫

1 2 1( )) ( ) ( ( )) ( ) ( ) ( ).T rT T rT ry xT e y T q T e x T e yτµ δ λ δ µ τ δ− −− + − + τ−

λ λ− = −∂ ∂ 1xm

Define the Lagrange multipliers to cause the coefficients of the respective variations on the states to

vanish, that is,

=- -1 1 1 /r H x 1λfx∂∂

, (A2)

2 2 2 /r H x =λ λ− = −∂ ∂ 2xm- - 2λhx∂∂

- 1µgx∂∂

, (A3)

=− 11 µµ r 2 /H y−∂ ∂ =- -ym 2λ h

y∂∂

- 1µ gy∂∂

, (A4)

with the boundary conditions

01 22 ( ) ( )

( )l

xλ τ λ τ

τ− + =0 (A5)

1 0 1sgn( ) 1( ) ( )

2e

y et TT s t T τµ δ δ τ + − +

= = + + − (A6)

2

2 ( ) xT sλ = .

Since x(0)=0 and y(

(A7)

τ )=0, (are fixed), their variations (0xδ and ( )yδ τ are zero thus ) Πδ becomes,

( )01 1 1 2 1( ) (

( )t rtH r l t l H y T e

x tsgn( ) 1)

2r T e

t

l t t Te dτ

δ δ −Π = + + − − + τ− −

=

⎛ ⎞+ − +⎜ ⎟⎝ ⎠

For the extreme, Πδ must be zero for arbitrary dτ this can only happen if

( )01 1 1 2 1( ) 2x t

sgn( ) 1( ) ( ) r T el t TH r l t l H y T e τ τδ − − + − ++ + − − + =0. (A8)

Considering the initial condition in the dual status, H1, H2 at τ , and (A5), (A8) becomes

02 1 22 ( )x

( )lC C λτ

− + + ( ( ))f x τ +0

1 1( )x τ( )lr lτ l+ − 2λ− ( ( ),0)h x τ 1µ− ( ( ),0)g xτ

( )

1sgn( ) 1( )

2r T et Ty T e τ τδ − − + − +

+ =0

Or denoting the left hand side by ( )Lτ , we have ( )Lτ =0.

Proof of Proposition 1:

The proposition follows directly from (12), implying it is never optimal to switch to a dual strategy as

ed. soon as the Rx is launch

Proof of Proposition 2:

The proposition follows directly from the assumption, from (12) and the continuity of L(τ ). Thus there is

τ (0, )T∈ such that L(τ )=0 and L(τ -)>0 but L(τ +)<0. Considering Remark 1, it is optimal to switch to

Proof of Pr

a dual strategy before the Rx patent expires.

oposition 3:

If L(T) 0≥ and there is no τ at which L(τ )=0, again from continuity of L(τ ), we obtain L(τ )>0, for

all τ (0, )T∈ , thus is opti al to stick entirely with the first optimization problem thus to switch only at

T.

m

6

2

ProProof of position 4:

The n follows directly from Theorem 1 and Remark 1 for the specific problem by substituting

the v

propositio

alues of 2 ( )λ τ and ( )µ τ (obtained from the solution of (18b)-(18c).

in Prescription Drug Consumers’ Spending :Computation of Savings

hen

x x ) rtx x y yyD p e dt−+ (A

isfies the first equation in (17b) but with ay set equal to 0 here to reflect only tho

erstwhile Rx drug users who switch to OTC drug use;

Let CS1 and CS2 denote originally Rx drug consumers’ spending on medication in Phases 1 and

respectively. T

rtCS xD p e dtτ

−= ∫ and (T

CS xD p= ∫1 1 2 20 τ

where y in (A9) sat

1xp , 2xp , are the Rx drug’s prices in the sing

and dual phases (assumed to be the same in our example); and py is the OTC version’s price. Substitutin

( ) ( , ) andx xCS F t D p e dtτ τ= ∫ x x y yCS G t D p H t D p e dtτ τ τ

(1a) and (2a), (A9) becomes

0

rtτ

−T

rt

τ

1 1 2 2( ) ( ( , ) ( , ) ) −= +∫ .

on Rx drug from allowing switch (SCSAS) will be

SCSAS(

Then, savings in consumer spending

τ )= CS1(T)-[ 1( )τCS + 2 ( )]τCS .

SCSAS depends on τ which in turn is affected by whether te = 3 or 0.

7

,

9)

se

2

le

g

28

29

30

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