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Essays in Industrial Organization: Market Performance

by

Mingxiao Ye

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Economics

University of Toronto

Copyright c© 2011 by Mingxiao Ye

Abstract

Essays in Industrial Organization: Market Performance

Mingxiao Ye

Doctor of Philosophy

Graduate Department of Economics

University of Toronto

2011

This thesis consists of three papers. Industries that motivated this analysis range

are exclusive clubs (Chapter 1) and pharmaceuticals (Chapters 2 and 3). A common

thread is the study of the strategic behavior of monopoly or monopoly-like firms and the

implications of such behavior.

Chapter 1 studies an “invitation only” strategy for a durable goods monopolist. “In-

vitation only” functions as a commitment device, enabling the extraction of more profit

than the conventional durable goods setting. In addition, the effectiveness of commit-

ment devices in profit-extraction can be compared: each commitment device is modeled

as an extra condition in the profit maximization of the general durable goods monopolist,

enabling straightforward comparisons across commitment devices.

Chapters 2 and 3 discuss the effect of patent protection on innovation in the pharma-

ceutical industry, in particular competition to produce drugs that follow-on from pioneer

drug discovery, and any feedback effects on pioneer innovation. Despite the conventional

notion, I show that longer patent protection may reduce or distort the incentives of in-

novation: with longer patents, the increased need of pioneer inventors in deterring the

production of follow-on drugs may translate to less profitability for the pioneer inventor.

Chapter 2 serves as a background and a literature review for Chapter 3. It explains

the multi-stage drug discovery process and the phenomenon of follow-on drugs; it reviews

strategic entry deterrence theories and summarizes the behavior of brand-name drug

ii

firms in deterring generic entry studied in the literature; it also reviews the optimal

patent length and breadth literature.

Chapter 3 presents several observed puzzles in the pharmaceutical industry and pro-

vides a unified explanation for these puzzles within a strategic entry deterrence model.

The central conclusion is that under some general conditions, longer patent life distorts

incentives for innovation and lowers research productivity: pioneer research is discour-

aged relative to follow-on research; inexpensive R&D projects are discouraged, and ceteris

paribus expensive projects are favored instead, especially those with large clinical trial

costs. Other predictions from the model accord with industry observations, including

mid-development cancellations of potential drugs for non-medical reasons and early de-

velopment of follow-on drugs in large markets.

iii

Acknowledgements

I am tremendously grateful to my supervisor Frank Mathewson, my committee mem-

bers Kenneth Corts and Carlos Serrano for their invaluable guidance and support. Work-

ing with them has been both a pleasure and a great opportunity for professional growth.

I’d especially like to thank Frank who is an inspiration with his insights, his kindness

and his wonderful sense of humor. I’d also especially like to thank Li Hao, without whom

Chapter 1 could not have been accomplished with the quality today, and whose generos-

ity with time I’m extremely grateful to. I would also like to thank the Department of

Economics, for the financial backing, research environment and administrative support.

Finally, I’d like to thank my friends and family: your love, support and trust are the

greatest assets in my life.

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Contents

1 Creating Artificial Demand 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Durable Goods Monopoly Benchmark . . . . . . . . . . . . . . . . . . . . 5

1.3 Model with “Invitation Only” Strategy . . . . . . . . . . . . . . . . . . . 9

1.4 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.1 Essential Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.2 Comparative Statics w.r.t. δ . . . . . . . . . . . . . . . . . . . . . 14

1.4.3 Profits for Each Stage . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5 Generalized Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.6 General Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.8 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.10 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.10.1 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . 29

1.10.2 Proof of lIN (x) ≥ lDG(x), ∀x ≥ 0 . . . . . . . . . . . . . . . . . . . 32

2 Pharmaceutical Innovation and Patent Protection 33

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2 The Drug Development Process . . . . . . . . . . . . . . . . . . . . . . . 34

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2.2.1 Competition in the Pharmaceutical Industry . . . . . . . . . . . . 37

2.3 The Controversy around Follow-on Drugs . . . . . . . . . . . . . . . . . . 38

2.4 Strategic Entry Deterrence . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4.1 Strategic Entry Deterrence Theories . . . . . . . . . . . . . . . . . 39

2.4.2 Deterrence of Generic Entry . . . . . . . . . . . . . . . . . . . . . 43

2.5 Optimal Patent Length and Breadth . . . . . . . . . . . . . . . . . . . . 48

2.5.1 The Effect of Patent Length on Innovation . . . . . . . . . . . . . 50

2.5.2 Contributions of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . 52

3 Arrested Development 53

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 The One-Stage Entry Game . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2.1 General Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2.2 Payoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2.3 SPEs of the One-Stage Game . . . . . . . . . . . . . . . . . . . . 66

3.2.4 (Dis)incentive Effect of Longer Patents . . . . . . . . . . . . . . . 68

3.3 The Two-Stage Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3.1 Payoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.3.2 SPEs of the Two-Stage Model . . . . . . . . . . . . . . . . . . . . 74

3.3.3 The Disincentive Effect Revisited . . . . . . . . . . . . . . . . . . 79

3.3.4 The Impact of the Disincentive Effect on Innovation Productivity 83

3.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.4.1 Multi-period Modeling . . . . . . . . . . . . . . . . . . . . . . . . 87

3.4.2 Predictions of the Two-Stage Model and Resolution of Paradoxes 88

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.6 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.6.1 Analysis of the disincentive effect in the one-stage model . . . . . 92

3.6.2 Analysis of the Two-stage model . . . . . . . . . . . . . . . . . . 94

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3.6.3 Analysis of the disincentive effect in the two-stage model . . . . . 100

Bibliography 103

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Chapter 1

Creating Artificial Demand:

Monopoly Profit Extraction

1.1 Introduction

“Invitation only” used to be a characteristic of exclusive clubs, like the National Arts

Club founded by “The New York Times” literary and art critic Charles de Kay, or of

cults or secretive organizations engaging in illegal activities, like the Ku Klux Klan. Even

today, the practice is still pervasive and it seems that exclusivity has hardly ever been

out of fashion. In recent years, the phenomenon has spread to the internet. Google’s

Gmail and Orkut, the former an email service and the latter a virtual community, are

both join-able by invitation only initially. INmobile.org runs an invitation only forum

of executives in the wireless industry. Vente-privee, a French company, holds invitation-

only closeout sales over the Internet. Off the web, many luxury brands, from clothing

to cosmetics, have annual or semiannual invitation only warehouse sales to deal with

overstock. Musicians hold invitation only shows. There are also invitation only auctions,

casinos, bars and so on.

It may seem apparent why weddings ceremonies and other events are invitation only,

1

Chapter 1. Creating Artificial Demand 2

if only to keep away potential disruptions. After all, no ultimate goal of any wedding

ceremony is to turn a profit (at least we’d like to think not). And it may be conjectured

that certain clubs are invitation only so as to maintain an elite status. But when it comes

to business, when the events or organizations are profit maximizing, it may be difficult to

explain the prevalence of the practice. Why should such a practice be chosen? While it

may convey an air of privilege and exclusivity, doesn’t it also severely limit the customer

base? Plus, does the sense of exclusivity actually lead to more willingness to pay on the

part of the “elites” that the practice caters to? If it is about catering to “elites”, then the

luxury brands’ discount sale is an oxymoron as the customers showing up are certainly

not there to feel exclusive and the joy of over-paying. Then, is it because the supply is

limited, that there are only 150 seats at King Arthur’s Round Table? But limited supply

is the problem, open market competition can always be employed to find a good price

and “invitation only” seems redundant. So what is the reason behind the practice of the

“invitation only” strategy?

To answer this question, let us investigate the following example.

A private city club primarily for business entertaining, the Metropolitan Club of

Chicago imposes an “invitation only” rule in membership. Its website claims the reason

is “to insure the perpetuation of a distinguished membership”, but this claim seems to

belie its true nature as a catering business. After all, a catering business is a business and

follows the basic rules. It is difficult to see what goal other than making profit should

exist. Why should “a distinguished membership” matter to this club? It’s not a club

intended for fostering a particular section of culture or a line of expertise. It’s not doing

this out of social responsibility. The membership criterion only matters if it affects profit.

Since there exists the good old simple way of setting prices very high so that only

“distinguished” citizens can afford it service, “invitation only” is not the unique, and

surely not the easiest way to sell. So how does imposing “invitation only” memberships

help in this profit-maximizing enterprize?


Further investigations indicate that this club is located on the 66th and 67th floors

of Sears Tower, the tallest building in North America and by height of “highest occupied

floor”, also the tallest in the world (at the time this paper is written). This has an

immediate implication: this particular catering business does not have many comparable

competitors because of the uniqueness in height. Indeed, for the purpose of investigating

the role of membership, it will be modeled as a monopoly.

This paper argues for a general model of such monopolies and illustrates the way in

which the “invitation only” strategy increases profit.

The result at first blush is counter-intuitive, as “invitation only” seems to further

limit the supply when available goods are rare, be they club memberships, seats in the

auction house, or space in a casino: if goods are only supplied to those furnished with an

invitation, many potential consumers who like the price are not eligible for purchasing.

And if availability is scarce in the first place, it makes sense to try to sell all the stock

instead of limiting the supply even further: the simple text-book monopolist model with

demand given by Q = α − βp (α, β > 0) suggests that the monopolist sell at the price

of α2β

and the quantity of α2, assuming marginal cost is zero. If production capacity is

limited, and less than α2

goods can be produced, then the more monopoly sells, the higher

the profit is. Thus “invitation only” seems to be reducing the profit intuitively.

But this is, of course, a false intuition. Limiting the supply does not always mean a

loss in profit, if the exclusivity brought about by the practice may materialize into a higher

propensity to pay by the individual customers–so much higher that it can make up for

the loss of sales volume and beyond. In this model, though “invitation only” membership

potentially limit the volume of sales, it enhances the expected value of membership itself

when a member has the privilege of deciding whom to invite. On the whole, profit is

increased by implicitly selling this privilege. The limitation in resource enables “invitation

only” strategy to create “artificial demand”: at a certain price level, there is more demand

than the case without such a strategy. This result can be interpreted as prestige creating


profit, and the purpose of the model is to explain how prestige can turn into profit when

a proper mechanism is employed.

The profit increase is the consequence of effective utilization of private information.

Observe that in a classical monopoly model, the monopolist is not able to distinguish

between consumers of different valuations, hence unable to extract as much profit as

when price discrimination is feasible. “Invitation only” strategy induces consumers of

different valuations to self-select into different groups. Thus the monopolist indirectly

implements the price discrimination.

In this model I assumed that invitations can be sold in a separate market, so that

the privilege of holding a membership (“a good”) can tangibly convert into money. This

assumption is for illustration purposes. The privilege is certainly not necessarily mon-

etary. The invitations need not be sold yet could still increase a member’s utility: it

may increase one’s influence in the club; it may also affect political balance, ideological

atmosphere or practical concerns; by picking a certain individual one increase the bond

with him/her, and may expect to receive reciprocity later on, etc.

A method of full extraction of profit, the Pacman strategy, is proposed by Bagnoli

et al. (1989), in which every period the monopolist posts a price equal to the highest

valuation among remaining consumers, and refuses to lower the price until the highest

valuation consumer buys it. The problem with this approach is that the number of

consumers has to be finite, so that each consumer expects her behavior can affect the

market. Moreover, when a small amount of noise is introduced, as in Levine & Pesendorfer

(1995), so that consumers’ valuations are not perfectly observable, then the approach fails

to work. Dudey (1995) argues that it is more reasonable to assume buyers should behave

as though they have no power in affecting future prices, even if the number of players

are finite.

My model avoids the above difficulties of Pacman strategy. It assumes a continuum of

consumers, yet by effectively decentralizing the market, “invitation only” still facilitates


the extraction of profit.

This model is also related to rationing in the durable goods monopoly literature. For

example, van Cayseele (1991) argues that the existence of low-valuation consumers is a

threat to the high-valuation ones when there is rationing, as high-valuation consumers

face the risk of being rationed when prices drop. Yet in his model, consumers can be

identified with their valuations. Spicer & Bernhardt (1997) verify that rationing with

finite consumers increases the profit, but their model requires the monopolist’s ability to

commit to destroy the good after any period when the planned amount is not sold. This

commitment may be difficult to come by in reality.

The paper is organized as follows: in Section 1.2 I discuss a durable goods monopolist

benchmark model for comparison; in Section 1.3 I discuss a two-stage “invitation only”

model, which is compared in Section 1.4 with the benchmark model; in Section 1.5, I

extend the original model to a generalized model for comparison across other means of

profit extraction of durable goods monopolists; Section 1.6 relaxes the assumption of

uniform distribution to include general distribution forms, and after further discussions,

concluding remarks are in section 1.9.

1.2 Durable Goods Monopoly Benchmark

In this section I am going to discuss a standard durable goods monopoly model, based

on those of Coase (1972) and Bulow (1982), for later comparisons.

There is a monopolist producer of a product and there is no time delay in production:

the monopolist is capable of instantly producing as many units of the product as she can

sell. There is a continuum of consumers. A consumer can purchase at most one unit of

the product. There are two stages, and at the beginning of each stage, the monopolist can

set a price that applies to this specific stage. The prices are observable to consumers, and

they make decisions after observing the price. The monopolist doesn’t know the private


valuation of any individual consumer, but knows the distribution of the valuations. The

time line is as follows:

• The monopolist sets the price p1 for the first stage;

• each consumer observes the price, and makes the decision whether to purchase a

unit of the good;

• the transactions occur and everyone observes the quantity sold in the first stage;

• the monopolist sets the second-stage price p2;

• everyone observes the second-stage price, and the consumers who did not purchase

in the first stage decide whether to purchase;

• second-stage transactions occur.

Denote the above game as DG, as in a typical durable goods monopolist setting.

Assume the private valuations of consumers follow an i.i.d uniform distribution be-

tween 0,1. The constant marginal cost of production is zero.

Under the above assumptions we have the following proposition:

Proposition 1. A weak sequential equilibrium of the two stage game DG is as follows:

1. The monopolist sets p1 = (2 − δ)2/2(4 − 3δ).

2. Letting q1 denote the amount sold in the first stage,

• the monopolist believes that consumers with valuations v ∈ [1−q1, 1] purchased

the goods in the first stage. She sets the price p2 = (1 − q1)/2. (In particular,

if q1 = 2(1 − δ)/(4 − 3δ), then p2 = (2 − δ)/2(4 − 3δ));

• consumers follow a cutoff strategy in the first stage. A consumer is willing to

purchase in the first stage if her valuation v ∈ [2p1/(2− δ), 1] and a consumer


who hasn’t previously purchased is willing to purchase in the second stage if

her valuation v ∈ [p2, 1].

As this is a standard version of durable goods monopolist model, the proof is omitted.

On the equilibrium path, the monopolist sells q1 = 2(1 − δ)/(4 − 3δ) at the price

p1 = (2−δ)2/2(4−3δ) in the first stage. In the second stage, she sells q2 = (2−δ)/2(4−3δ)

at p2 = (2− δ)/2(4− 3δ), a lower price than in the first stage. The consumers self-select

into three groups. The consumers with high valuations form the first group, who purchase

in the first stage. The middle valuation consumers form the second, purchasing in the

second stage. The low valuation consumers do not purchase in either of the stages. I’ll

briefly explain how the values of the variables are obtained in equilibrium.

Assuming rational expectations, in equilibrium consumers who purchase in the first

stage must prefer it to purchasing in the second stage, and vice versa. There must exist

a valuation level c that a consumer with such a valuation would be indifferent between

purchasing at the first and the second stage. The arbitrage condition for this consumer

specifies

c − p1 = δ(c − p2) (1.2.1)

In the second stage, the monopolist sets the price p2 to maximize the second-stage profit,

given his belief of the distribution of the private valuations of the consumers left from

the first stage. In equilibrium, the belief is correct and the consumers left from the first

stage have valuations uniformly distributed between 0, c.

maxp2

p2(c − p2) (1.2.2)

The first order condition gives

p2 = c/2 (1.2.3)

The second order condition is satisfied.


In the first stage, knowing that the monopolist is going to maximize the second-stage

profit, consumers rationally expect what the second stage price is going to be, on learning

the first-stage price, and c is determined in this way.

In the first stage, the monopolist chooses the prices knowing their impact on c.

maxp1,p2

p1(1 − c) + δp2(c − p2) (1.2.4)

s.t. (1.2.3) and (1.2.1)

Solving, we have

cDG =2 − δ

4 − 3δ(1.2.5)

p2,DG =2 − δ

2(4 − 3δ)(1.2.6)

p1,DG =(2 − δ)2

2(4 − 3δ)(1.2.7)

The total profit from both stages is

ΠDG =(2 − δ)2

4(4 − 3δ)(1.2.8)

The durable goods monopolist faces the problem of not being able to commit to the

second-stage price at the beginning of the game. She has to maximize the second-stage

profit, once the time moves to the second stage. The consumers rationally expect that

and they respond by being less willing to purchase in the first stage, since the price is

higher than in the second stage. If instead she could commit to the second-stage price,

she could have set both prices at 1/2, and everyone with valuation v ∈ [1/2, 1] would

have bought in the first stage. The profit would be 1/4, which is equal to the static


monopoly profits, and strictly greater than ΠDG = 2−δ2

4(4−3δ)when δ < 1. As predicted by

the Coase conjecture (Coase 1972), the durable goods monopolist is unable to achieve

static monopoly profits.1

A monopolist’s nature of chasing profit is thus understood by the consumers, and

they won’t believe her promise of not doing so in the second stage. But sometimes, there

are ways of making plausible promises, so that the monopolist may benefit from them.

For example, a sculptor may destroy the mould her sculptures are made from. But these

involve special circumstances, and not all monopolists are able to do something similar.

In the next section, I am going to explain how an “invitation only” strategy works in

much more general circumstances.

1.3 Model with “Invitation Only” Strategy

In this model, the monopolist sells in the first stage as usual, but in the second stage,

she commits to sell only to those who hold an invitation. Any consumer who purchases a

unit of good in the first stage automatically gains the right to send out invitations. The

number of invitations one can send out is unlimited, as long as one can find a recipient.

The time line is as follows.

• The monopolist sets the price p1 for the first stage;

• each consumer observes the price, and decides whether she wishes to purchase a

unit of the good, and the trading occurs;

• everyone observes the quantity sold in the first stage;

• the monopolist sets the second-stage price p2;

1Ausubel & Deneckere (1989) show that equilibria exist in which the durable goods monopolist earnsapproximately the static monopoly profits if the monopolist can commit to a price sequence, and thetime interval between the successive price offers approaches zero.


• everyone observes the second-stage price;

• the consumers who did not purchase in the first stage decide whether to enter the

invitation market (those who wish to purchase in the second stage are henceforth

called “second-stage consumers”);

• each second-stage consumer is randomly matched pairwise with some first-stage

consumer in the invitation market;

• each first-stage consumer makes a take-it-or-leave-it offer to his partner(s);

• each second-stage consumer responds with yes or no;

• those who obtain an invitation purchase in the second-stage market.

Denote the above game IN . As in DG, assume the private valuations of consumers

follow an i.i.d uniform between 0,1. The constant marginal cost of production is zero.

Assume the monopolist is able to commit to sell only to invitation holders in the second

stage.

Notice that because no one knows the valuation of the person he’s matched with in the

invitation market, the take-it-or-leave-it offer must be the maximizer of expected payoff

in selling an invitation. Thus it’s the same for every seller and every buyer. Denote this

offer (price of a invitation) as k.

In the IN game, there is also a critical consumer for whom the arbitrage condition

holds. Denote his valuation as c. This person is indifferent between purchasing in the first

and second stage. If he purchases in the first stage, he obtains his valuation of the good

c, and the prospect of selling invitations. Only those with valuation v ≥ p2 + k would

purchase an invitation, since anyone with a lower valuation makes a loss by doing that.

Since the first stage cutoff is c, there is mass c − p2 − k buyers of invitations, and mass

1− c of sellers. Thus each seller can expect to sell (c− p2 −k)/(1− c) invitations at price

k each. Discounted because of the time preference, the prospect of selling invitations


amounts to δk(c − p2 − k)/(1 − c). If he purchases in the second stage, he obtains a

discounted value of the good minus the prices he pays in both the invitation and goods

market, which in total amounts to δ(c − p2 − k).

Thus the arbitrage condition for the critical consumer specifies

c − p1 + δkc − p2 − k

1 − c= δ(c − p2 − k) (1.3.1)

Proposition 2. A weak sequential equilibrium of the two stage game IN is as follows:

1. The monopolist sets p1 = (4 − δ)2(16 − 9δ)/4(16 − 7δ)(8 − 5δ).

2. Letting q1 denote the amount sold in the first stage,

• the monopolist believes that consumers with valuations v ∈ [1−q1, 1] purchased

the goods in the first stage. She sets the price p2 = (1 − q1)/2. (In particular,

if q1 = (8 − 5δ)/(16 − 7δ), then p2 = (4 − δ)/(16 − 7δ));

• consumers follow a cutoff strategy in the first stage. A consumer is willing

to purchase in the first stage if her valuation v ∈ [c, 1] where c satisfies p1 =

c+δc2/16(1−c)−δc/4. A consumer who hasn’t previously purchased is willing

to purchase in the second stage if his valuation v ∈ [p2 + k, 1], where k is the

price of invitation.

3. In the invitation market, any first-stage consumer believes that consumers with

valuations v ∈ [1− q1, 1] purchased the goods in the first stage, and he proposes the

price k = (1 − q1 − p2)/2. A second-stage consumer accepts the offer if and only if

his private valuation is no less than p2 + k.

The proof is in the Appendix.

By the proposition, on the equilibrium path, p1 = (4−δ)2(16−9δ)/4(16−7δ)(8−5δ),

first-stage consumers are those with v ∈ [(8−2δ)/(16−7δ), 1] and second-stage consumers


are those with v ∈ [3(8−2δ)/4(16−7δ), (8−2δ)/(16−7δ)]. The price the monopolist sets

in the first stage is higher than that of the second stage. And similar to that of the game

DG, the consumers self-select into three groups. The consumers with high valuations

form the first group, who purchase in the first stage. The middle valuation consumers

form the second, obtaining an invitation and then purchase in the second stage. The low

valuation consumers do not purchase in either of the stages.

1.4 Comparative Statics

1.4.1 Essential Variables

Denote the profit of durable goods monopolist as ΠDG, and the profit of invitation only

strategy as ΠIN . I use similar subscripts for other variables. To facilitate comparison, I

denote by l the consumer with the lowest valuation among consumers who purchase the

goods (or, the last consumer). In both games, the last consumer is the consumer with

the lowest valuation who purchases in the second stage and breaks even with a net utility

of zero. In DG, this consumer’s valuation is simply the second-stage price, as he breaks

even. In IN , this consumer’s valuation is higher than the second-stage price by the price

of the invitation, in which case he also breaks even.

All the following comparisons are made with the same δ.

1. ΠDG ≤ ΠIN , = when δ = 1;

Proof. For ΠDG > ΠIN , we need (4 − δ)2/(16 − 7δ) > (2 − δ)2/(4 − 3δ) which

simplifies to δ(δ − 3)(δ − 1) > 0. It holds whenever δ < 1.

The profit for the monopolist is higher when she employs the “invitation only”

strategy. Why this is the case will be explained in detail in later sections, when I

compare the games on a general background.


2. cDG ≥ cIN , = when δ = 0. The same for p2;

Proof. For cDG ≥ cIN , we need (2−δ)/(4−3δ) ≥ (8−2δ)/(16−7δ) which simplifies

to δ(δ + 2) ≥ 0. It holds for p2 as in both models, p2 = c/2.

The critical consumer has a lower valuation in IN . Intuitively, compared with

waiting till the second stage, paying for both the invitation and the good, a high

valuation consumer in IN has more to gain to purchase immediately at a higher

price.

3. lDG ≤ lIN , = when δ = 1;

Proof. As lDG = p2,DG = (2 − δ)/2(4 − 3δ) and lIN = p2,IN + k = 3cIN/4 =

3(8−2δ)/4(16−7δ), for lDG ≤ lIN we need (2−δ)/2(4−3δ) ≤ 3(8−2δ)/4(16−7δ).

It simplifies to (δ − 8)(δ − 1) ≥ 0, which holds as long as δ ≤ 1.

Since the last consumer has a higher valuation in IN , the total quantity sold in

IN is smaller. Intuitively, “invitation only” strategy limits the effective supply by

creating friction in the market.

4. p1,DG ≤ p1,IN , = when δ = 0.

Proof. Notice that when δ = 0, both p1,DG = (2−δ)2

2(4−3δ)and p1,IN = (4−δ)2(16−9δ)

4(16−7δ)(8−5δ)equal

to 1/2. When δ = 1, p1,DG = 1/2, and p1,IN = 7/12. The derivative with respect

to δ is [K1](3δ−2) in DG, and in IN is [K2][(16−9δ)(136−70δ)− (68−27δ)(16−

7δ)(8−5δ)] = [K2](315δ2−1188δ+1152) ≥ 0, = when δ = 0. K1, K2 are constants.

Thus p1,DG < p1,IN whenever δ > 0. See diagram.


1

0.4

0.5

0.6

0.7

P1P1, DG P1, IN

DG

IN

The first-stage price in IN is higher. It embodies not only the value of the good,

but also that of the invitations.

1.4.2 Comparative Statics w.r.t. δ

How do the strategies of the monopolist and the consumers change, when they become

more patient, such that selling/purchasing in the first stage enjoys less of a utility ad-

vantage than selling/purchasing in the second? In the following tables I list how the

essential variables and change in δ, and the values they take when δ goes to the limits

on both ends.

DG c p2 p1 Π l

Value 2−δ4−3δ

c/2 (2−δ)2

2(4−3δ)(2−δ)2

4(4−3δ)c/2

In δ ↑ ↑↑ when δ ≥ 2/3;

↓ when δ < 2/3

↑ when δ ≥ 2/3;

↓ when δ < 2/3↑

δ → 0 1/2 1/4 1/2 1/4 1/4

δ → 1 1 1/2 1/2 1/4 1/2

When δ increases, ie. consumers become more patient, the critical consumer’s valu-

ation increases. For a high valuation consumer, purchasing immediately used to make


much of a difference from waiting, as his high valuation generates a big loss in waiting.

But with more patience, the loss is smaller, and even a high valuation consumer can

afford to wait. The comparative statics also indicates that when the patience level is low

(< 2/3), the optimal first-stage price is lower with more patience, as a high price causing

more consumers to wait is more damaging than the loss in immediate profit. But when

the patience level is high, the reverse is true.

In the limit, when everyone becomes extremely impatient, only the first stage mat-

ters. This becomes a standard one period monopoly problem. When everyone becomes

extremely patient, selling and purchasing in the first and the second stage make no differ-

ence, and the first and second-stage price is equalized, with the setting again becoming

a standard one period monopoly problem.

IN c p2 p1 Π l k

Value 8−2δ16−7δ

c/2 (4−δ)2(16−9δ)4(16−7δ)(8−5δ)

(4−δ)2

4(16−7δ)3c/4 c/4

In δ ↑ ↑ ↑↑ when δ ≥ 4/7;

↓ when δ < 4/7↑ ↑

δ → 0 1/2 1/4 1/2 1/4 3/8 1/8

δ → 1 2/3 1/3 7/12 1/4 1/2 1/6

When consumers become more patient, the critical consumer’s valuation increases for

the same reason as in DG. Prices in the first stage also increase, as the benefit of having

a higher first-stage price outstrips the loss incurred when consumers wait for the second

stage. Price in the second stage also increases, as the critical consumer’s valuation is

higher.

In the limit, when everyone becomes extremely impatient, it becomes a standard one

period monopoly problem since only the first stage matters. When everyone becomes


extremely patient, purchasing in the first and the second stage make no difference, taking

the invitations into consideration. The profit level is again at par with that of the

standard monopoly problem.

1.4.3 Profits for Each Stage

Realistically, the economic agents are neither completely impatient nor completely pa-

tient. From now on we only consider 0 < δ < 1, so that all the inequalities in section

1.4.1 are strict.

Is IN making more profit for the monopolist because in both stages it does better

than DG?

We have the following results:

1. Π1,DG < Π1,IN

Proof. Since cDG > cIN and p1,DG < p1,IN , there are more people purchasing at a

higher price in the IN model. It follows the the profit in the first stage is higher

in IN .

This is easy to understand since first-stage consumers are motivated by the prospect

of selling the invitation in the IN model, and naturally IN model does better than

DG in the first stage. But this is at the cost of making less profit in the second

stage, because invitation serves as a barrier to enter the second-stage goods market,

and less people show up. Also, the second-stage price is lower in IN since c is lower.

2. Π2,DG > Π2,IN

Proof. Π2,DG = δ(4 − δ)2/2(16 − 7δ)2 and Π2,IN = δ(2 − δ)2/4(4 − 3δ)2


Still, the total effect for IN is that the profit gain in the first period is greater than

the profit loss in the second period. To understand why this is the case, we shall look at

a generalized problem.

1.5 Generalized Problem

Consider a family of models describing the monopolist’s problem with asymmetric infor-

mation. Each consumer is to purchase at most one unit of good. Consumers have private

valuations that are unknown to the monopolist, except the continuous non-atomic dis-

tribution. Thus the monopolist can not treat the consumers differentially except by

charging different prices in different stages. And assume the model doesn’t specify con-

sumer’s behavior except that they behave rationally. In such models, the following results

must hold:

1. From each consumer that purchases in the same stage, the same amount of profit is

extracted from each. This is because the profit is simply the price minus the cost,

and both are equal among consumers from the same stage. Thus in each stage,

each consumer’s surplus increases one-to-one with her private valuation.

Denote the amount of profit the monopolist extracts from each first stage consumer

as e1 and from a second-stage consumer as δe2 (already discounted).

2. There exists a critical consumer whose expected surplus is the same in both stages,

as long as there exists at least one consumer at each stage. This follows from the

continuity of the problem and the continuity of valuations.

Without a critical consumer, there is discontinuity in expected surplus between the

two stages. Suppose consumer A is the last consumer in the first stage. Then for

consumer A to be optimizing, it must be strictly worse for A to go to second stage

instead. Denote the difference as n > 0, then S1(vA) − S2(vA) = n where S1 and


S2 denote surplus from the first and second stage respectively, and are continuous

functions. But then consumer B who’s valuation is arbitrarily close to A and who

purchases in the second stage could have switched to the first stage and be better

off, since 0 < S1(vB) − S2(vB) < n due to the continuity of surplus functions.

Denote the critical consumer’s valuation as c.

3. The surplus of the last consumer is zero. This follows from consumers’ free entry

into the markets and the continuity of valuations.

Denote the last consumer’s valuation as l.

Now we can formalize the problem according to the above.

The monopolist’s problem is

maxe1,e2,l,c

e1(1 − c) + δe2(c − l) (1.5.1)

s.t. c − e1 = δ(c − e2) (critical consumer) (1.5.2)

l = e2 (last consumer) (1.5.3)

Combine the restrictions with the objective, we have

maxl,c

[c − δ(c − l)](1 − c) + δl(c − l) (1.5.4)

The objective function and the contour curves are shown below:


The Objective

0

0.25

0.5

0.75

1 0

0.25

0.5

0.75

1

0

0.1

0.2

0

0.25

0.5

0.75

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Contour Graph of the Objective

There may be other restrictions to the maximization that’s specific to the individual

model.


For example, in the DG model, l and c have an extra relation given by the second-

stage maximization of the monopolist. The two variables have one degree of freedom

instead of two. This is because the monopolist has to choose a maximizing price in the

second stage. He can not commit not to do so. As described before, that would require

he chooses p2 to maximize second-stage profit, given his belief of c, (c isn’t observable to

him, instead he deduces it correctly in equilibrium.)

maxp2 p2(c − p2)

But p2 is simply l, so the first order condition would give

l = c/2 (1.5.5)

Similarly, in the IN model, there is an extra relation between l and c. It also originates

from the monopolist’s inability to commit to act non-optimally in the second stage, except

this time he is able to commit to sell only to those with invitations.

As before, let sellers of invitations optimally choose price k, and let the monopolist

optimally choose p2. We have p2 = c/2 and k = c/4. Since l = p2 + k, we have

l = 3c/4 (1.5.6)

To see that the DG model corresponds to the general problem plus (1.5.5), notice

that for the DG model, e1 = p1 and δe2 = δp2. The arbitrage condition c−p1 = δ(c−p2)

in terms of e1 and e2 is c− e1 = δ (c − e2). The objective maxp1,p2,c p1(1− c)+ δp2(c− p2)

is translated into maxe1,e2,l,c e1 (1 − c) + δe2 (c − l). Then the DG monopolist’s problem

corresponds exactly to the one in the general problem, except that since the monopolist

cannot commit to the second-stage price, there is an extra condition (1.5.5).

For the IN model, e1 = p1 − δk c−p2−k1−c

, k = c−p2

2, l = p2 + k, and δe2 = δl. The

arbitrage condition c − p1 + δk c−p2−k1−c

= δ(c − p2 − k) expressed in terms of e1 and

e2 is c − e1 = δ (c − e2). The objective p1 (1 − c) + δp2 (c − p2 − k) is translated into


e1 (1 − c) + δe2 (c − l). Then the IN monopolist’s problem also corresponds exactly to

the one in the general problem, except for the extra condition (1.5.6).

Without the conditions (1.5.5) from DG and (1.5.6) from IN , the solution to the

problem is (c, l) = (1/2, 1/2). Denote the objective e1(1 − c) + δe2(c − l) as H. Then

the derivative of H with respect to c is Hc = (1 − δ)(1 − 2c), to l is Hl = δ(1 − 2l).

Recall that δ < 1. Notice Hc is positive when c < 1/2 and negative when c > 1/2, thus

decreasing when the distance with 1/2 increases. Same is true for Hl.

This becomes apparent in the following graph. The profit level curves are elliptical,

and as any point moves further away from the maximum point (1/2, 1/2) the lower the

profit is. The extra conditions for the IN and DG model, (1.5.5) and (1.5.6), are both

straight lines crossing the origin, with IN curve having a greater slope. Apparently the

IN model should achieve a higher profit.

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1DG and IN constraints


1.6 General Distribution

The profit relation between DG and IN holds even if the distribution isn’t uniform.

Assume the distribution is non-atomic on [0, 1] with cdf F (∙) and pdf f (∙). For simplicity,

assume the objectives are concave, and as can be seen later on in the paper, this can be

guaranteed by the following two assumptions:

1. (SOCDG): −2f(l) − lf ′(l) ≤ 0, ∀l

2. d2[F (c)−F (l)f(l)

]/dl2 ≥ 0, ∀l, c

Notice that the uniform distribution satisfies both assumptions.

As before, assume the monopolist can extract e1 from a first stage consumer, and δe2

from a second-stage consumer (already discounted). Let the critical consumer’s valuation

be c, and the last consumer’s be l.

The monopolist’s problem is

maxe1,e2,l,c

e1[1 − F (c)] + δe2[F (c) − F (l)] (1.6.1)

s.t. c − e1 = δ(c − e2)(critical consumer) (1.6.2)

l = e2 (last consumer) (1.6.3)

Combining the constraints, the problem can be written as

maxl,c

[c − δ(c − l)][1 − F (c)] + δl[F (c) − F (l)] (1.6.4)

As before, for the DG model, there is an extra condition imposed by the maximizing

behavior of the monopolist in the second stage.

The second-stage maximization maxp2 p2[F (c) − F (p2)] in terms of l and c is simply

maxl

l[F (c) − F (l)] (1.6.5)



F (c) − F (l) − lf(l) = 0 (FOCDG)

For simplicity, assume the objective is concave:

−2f(l) − lf ′(l) ≤ 0, ∀l (SOCDG)

For the DG model, e1 = p1 and δe2 = δp2. The arbitrage condition c− p1 = δ(c− p2)

in terms of e1 and e2 is c − e1 = δ (c − e2). The objective maxp1,p2,c p1[1 − F (c)] +

δp2[F (c) − F (p2)] is translated into maxe1,e2,l,c e1[1 − F (c)] + δe2[F (c) − F (l)]. Then the

DG monopolist’s problem corresponds to the one in the general problem, with the extra

condition FOCDG.

For the IN model, the extra condition as usual is imposed by the two steps of maxi-

mization in the second stage.

1. First, the sellers of invitations, on learning the second-stage price, set up the price

for invitations by

maxk

k[F (c) − F (k + p2)]/[1 − F (c)] (1.6.6)


F (c) − F (k + p2) − kf(k + p2) = 0 (1.6.7)

For concavity of the objective, we need

− 2f(k + p2) − kf ′(k + p2) ≤ 0, ∀k (1.6.8)

Notice p2 is given at the time the maximizing k is chosen through (1.6.6).


Since in the second stage, the last consumer is the one with valuation l = k + p2,

the above first and second order conditions in terms of c, l and k is

F (c) − F (l) − kf(l) = 0 (FOCk)

and

−2f(l) − kf ′(l) ≤ 0, ∀l (SOCk)

respectively, where k = l − p2.

Notice when SOCDG holds, SOCk constantly holds: when (SOCDG) holds, −f ′(l) ≤

2f(l)/l, so the LHS of SOCk is −2f(l)− (l−p2)f′(l) ≤ −2f(l)+ (l−p2)[2f(l)]/l =

−2p2f(l)/l ≤ 0. Thus SOCk holds under SOCDG as long as p2 ≥ 0.

2. Second, the monopolist sets the second-stage price by maxp2 p2[F (c) − F (k + p2)]

subject to FOCk, which is simply

maxl

(l − k)[F (c) − F (l)] (1.6.9)

s.t. (FOCk)


(1 − k′)[F (c) − F (l)] − (l − k)f(l) = 0 (FOCIN )

where k′ ≡ dk/dl = −[f(l) + kf ′(l)]/f(l) is given by implicit differentiation of

FOCk. And for simplicity, we need some assumption to guarantee that ( l−k)[F (c)−

F (l)] is concave. This can be achieved by assuming k′′ ≡ d2k/dl2 = [F (c)−F (l)f(l)

]′′ ≥

0, ∀l, given any c: the second order derivative of k[F (c) − F (l)] with respect to

l is k′′[F (c) − F (l)] − 2k′f(l) − kf ′(l) = k′′[F (c) − F (l)] + 2f(l) + kf ′(l) since

k′ = −[f(l) + kf ′(l)]/f(l). Because of SOCk, k′′[F (c) − F (l)] + 2f(l) + kf ′(l) ≥


k′′[F (c)−F (l)]. Since k′′ ≥ 0 and c ≥ l, the second order derivative of k[F (c)−F (l)]

is non-negative. Thus k[F (c)− F (l)] is convex in l. Since l[F (c)− F (l)] is concave

in l by SOCDG, (l − k)[F (c) − F (l)] is concave in l.

For the IN model, e1 = p1−δk[F (c)−F (k+p2)]/[1−F (c)], and δe2 = δl = δ(p2 +k).

The arbitrage condition c−p1+δk[F (c)−F (k+p2)]/[1−F (c)] = δ(c−p2−k) expressed in

terms of e1 and e2 is c−e1 = δ (c − e2). The objective is e1 (1 − c)+δe2 (c − l). Then the

monopolist’s problem in IN also corresponds to the one in the general problem, except

for the extra condition FOCIN .

Without the extra restriction from DG or IN , the solution to the monopolist’s prob-

lem (1.6.4) is (c, l) = (a, a) where a is the solution to the equation

1 − F (a) − af(a) = 0 (1.6.10)

assuming the second order conditions for concavity hold: Denote the objective as J , then

both ∂2J/∂c2 = (1−δ)[−2f(c)−cf ′(c)] ≤ 0 and ∂2J/∂l2 = δ[−2f(l)−lf ′(l)] ≤ 0 because

of SOCDG, which is already assumed to hold; and the cross derivative is zero. Since the

concavity of (1.6.4) is assumed to hold, notice Jc is positive when c < a and negative

when c > a, thus decreasing when the distance with a increases. Same is true for Jl.

Like the general distribution case, the profit level curves are closed curves around

the maximal point (a, a). The extra conditions for DG and IN models, FOCDG and

FOCIN , turn out to be lines crossing the origin but below (a, a), and the IN curve is

constantly “above” the DG curve.

To see FOCDG is below (a, a), notice that the LHS of FOCDG is a decreasing function

of l because of SOCDG, and at (a, a) the LHS is negative according to the definition of

a in (1.6.10). It follows that from the point (a, a), holding c = a constant, decreasing l

can reach a point on FOCDG.


To see FOCIN is below (a, a), notice that the LHS of FOCIN is a decreasing function

of l because of SOCIN , and at (a, a) the LHS is negative according to the definition of a

in (1.6.10): LHS of FOCIN is −(l − k)f(l) which is negative.

To prove IN curve is constantly “above” the DG curve, we need to prove l′IN (c) ≥

l′DG(c), ∀x ≥ 0, where lIN (c) is the function defined by the implicit function FOCDG;

lDG(c) is the function defined by FOCIN .

Recall FOCDG, when c = 0, −F (l) − lf(l) = 0. But F (.) and f(.) are nonnegative,

so l = 0. Similarly, according to FOCIN , when c = 0, l = k = 0. Thus we only need to

prove lIN (x) ≥ lDG(x), ∀x ≥ 0.

The proof of lIN (x) ≥ lDG(x), ∀x ≥ 0 can be found in the Appendix.

The graph below shows the relation between the constrained maximization of DG

and IN .

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1General Distribution


1.7 Discussion

The generalized problem discussed before provides not only the upper bound of profit in

any monopolist’s problem facing incomplete information, but also a quick way to evaluate

the profit potential of various proposals.

Spicer & Bernhardt (1997) proposes that in a discrete model, probabilistic rationing

doesn’t improve a durable goods monopolist’s profit above that of a static monopolist’s.

By looking at the generalized problem it is immediately clear why it should be so and

why this result should also hold for continuous models.

Apparently, a durable goods monopolist’s model with rationing falls into the general

problem, with the extra condition that the monopolist has to maximize the second period

profit without breaking the ration. A maximization with more constraints is certainly

not going to do better than with less.

The problem is, can it do better than the standard DG model? The answer is possible.

If the ration is such that it’s right at the static monopolist’s maximizing amount. Then

it follows that after the first-stage sales, the monopolist optimally sells off the rest of the

goods in the second stage. Then the extra condition is simply l = 1 − q where q is the

ration level. It follows that p2 = l = 1/2. As long as c = 1/2, the monopolist achieves

the static monopolist’s profit. Committing to ration is equivalent to committing to the

second-stage price.

1.8 Extensions

This model focuses on the setting of two stages as it sufficiently illustrates the theme of

the paper, without too much complication and mathematical clutter. But future research

can be done to related models with more than two stages, which may be sufficiently closer

to real world events to justify the amount of technicality.

Some aspects of the model setting may be changed to generate interesting questions.


In the IN model, I assumed that first-stage consumers have no knowledge about the

second-stage consumers’ private valuation. This is not often the case if we think about

the real life scenarios, in which current members invite their friends, whose tastes may

more or less be familiar to them. The point of the IN model is, even without the private

information which the monopolist can potentially exploit, she can still be better off with

the “invitation only” strategy. What if in addition, the current members have certain

knowledge about the ones they may invite? Does that generate extra profit? And is

“invitation only” going to efficiently utilize this knowledge?

Even if we keep the assumption of current members having no knowledge about

potential consumers, we might be interested in whether a drastically different result may

arise if the matching between current and potential members is not random. Or, if each

current member can only send out a limited number of invitations, does that generate so

much friction in the market that “invitation only” may not work?

Additional research may be done on combining different strategies, for example, ra-

tioning and “invitation only”. Or, since we’ve studied a family of generalized problems,

new strategies may be proposed in cases that “invitation only” does not work–for exam-

ple, when the monopolist is not able to commit to sell only to the invitation holders.

1.9 Conclusion

Under the standard durable goods monopolist model setting, the monopolist knows the

continuous non-atomic distribution of consumers’ valuations but not any individual’s

private valuation. If each consumer may purchase at most one unit of good, and if the

monopolist can commit to sell only to invitation holders in later periods, “invitation only”

is one way to extract more profit for the monopolist. The reason for the higher profit,

as the study on the generalized problem clarifies, is that “invitation only” substitutes a

standard condition on profit maximization with a more relaxed condition. Potentially,


there exists other strategies that can increase the profit level above the standard durable

goods monopolist level, and this paper provided guidelines to find such feasible strategies:

as long as the monopolist can commit to certain behavior like selling only to invitation

holders in the second stage, which implies more relaxed restrictions imposed by the

monopolist’s maximizing behavior, the monopolist is able to extract more profit. This

conclusion may be applied to predicting the effects of relevant business strategies: instead

of setting up a full model for analysis, one can compare the restricting conditions inherent

in each strategy, conditions on the private valuations of the “critical consumer” and the

“last consumer”, and the more relaxed condition corresponds to a higher profit.

1.10 Appendices

1.10.1 Proof of Proposition 2

Proof. The monopolist makes the choice of p1 at the beginning of the game, and p2 after

every history of the realization of q1. Each consumer decides whether to show up in

the first stage, second stage and invitation markets, and what to offer in the invitation

market if he has purchased in the first stage, and what offers to accept in the invitation

market if he wishes to purchase in the second stage.

1. Sequential rationality of consumers.

• In the invitation market, given the beliefs about the cutoff c of consumers in

the first stage, a seller of invitation should set the price k to maximize the

expected payoff:

maxk

k(c − p2 − k)/(c − p2) (1.10.1)

Then

k = (c − p2)/2 (1.10.2)


But in his belief c = 1 − q1, thus k = (1 − q1 − p2)/2.

A second-stage consumer purchases an invitation if and only if it generates

non-negative surplus for him in the second stage.

• For the decision about entering the first-stage market, the cutoff strategy in

the proposition is simply the arbitrage condition with the conditions on k and

p2 taken into account:

Recall the arbitrage condition (1.3.1) specifies c−p1+δk c−p2−k1−c

= δ(c−p2−k).

From k = (1 − q1 − p2)/2, p2 = (1 − q1)/2 and c = 1 − q1, we have k = c/4

and p2 = c/2. Thus the arbitrage condition becomes c− p1 + δc c/161−c

= δ(c/4),

which simplifies to

p1 = c + δc2/16(1 − c) − δc/4 (1.10.3)

If a consumer’s private valuation is lower than c, then he would be better off

to wait until the second stage or refrain from purchasing at all, while anyone

with valuation above c should purchase in the first stage instead of waiting.

• For the decision of entering the second-stage market, notice that anyone with

valuation v ∈ [p2 + k, 1] stands to gain non-negative surplus by entering the

second-stage market, while anyone with valuation lower than that stands to

lose.

2. Sequential rationality of the monopolist.

• In the second stage, when price in the first stage was set at p1 and q1 goods

have been sold, the monopolist sets the optimal price p2 given his belief of the

distribution of the private valuations of the consumers left from the first stage,

and knowing that whatever p2 she sets, in equilibrium k = (1 − q1 − p2)/2.

Then the consumers who are going to purchase in the second stage are those

with valuations v ∈ [p2 + k, 1].


The optimal p2 is solved by

maxp2

p2[1 − q1 − p2 − (1 − q1 − p2)/2] (1.10.4)

It gives p2 = (1 − q1)/2.

• In the first stage, the monopolist chooses the optimal prices anticipating the

consumers’ cutoff strategy after p1 is announced. In equilibrium, the antici-

pation is correct and the consumers left from the first stage have valuations

uniformly distributed between 0, c. Thus

p2 = c/2 (1.10.5)

The first-stage optimization gives

maxp1,p2,c

p1(1 − c) + δp2(c − p2 − k) (1.10.6)

s.t. (1.10.5) and (1.10.3)

Solving, we have

cIN =8 − 2δ

16 − 7δ(1.10.7)

p2,IN =4 − δ

16 − 7δ(1.10.8)

p1,IN =(4 − δ)2(16 − 9δ)

4(16 − 7δ)(8 − 5δ)(1.10.9)

ΠIN =(4 − δ)2

4(16 − 7δ)(1.10.10)

3. Now I confirm that the monopolist and consumers uses Bayesian updating on the

equilibrium path. In equilibrium, p1 = (4−δ)2(16−9δ)4(16−7δ)(8−5δ)

, the first-stage consumers are


those with valuations v ∈ [(8 − 2δ)/(16 − 7δ), 1], and the amount of goods sold is

q1 = 1− (8− 2δ)/(16− 7δ). The monopolist and consumers correctly update their

belief by believing the first-stage consumers’ valuations are uniform on [1 − q1, 1],

which is [(8 − 2δ)/(16 − 7δ), 1].

1.10.2 Proof of lIN(x) ≥ lDG(x), ∀x ≥ 0

Proof. Notice FOCDG is acquired by (1.6.5), while FOCIN is by (1.6.9). Since both

objectives are concave, for lIN (x) > lDG(x), ∀x ≥ 0, we only need to prove that ∀c, at the

l that maximizes (1.6.5) (henceforth l∗DG), the slope of (1.6.9) would be non-negative.

At l∗DG, the slope of (1.6.9) is

0 −dk

dl|l=l∗DG

[F (c) − F (l∗DG)] + kf(l∗DG) (1.10.11)

By implicit differentiation of FOCk, dk/dl = −[f(l) + kf ′(l)]/f(l). Evaluating at

l∗DG, plugging into (1.10.11) gives

2 + f ′(l∗DG)[F (c) − F (l∗DG)]/f 2(l∗DG) (1.10.12)

Since F (c) − F (l∗DG) = l ∗DG f(l∗DG) according to FOCDG, (1.10.12) is simplified

to 2 + f ′(l∗DG)l ∗DG f(l∗DG)/f 2(l∗DG) = 2 + f ′(l∗DG)l ∗DG /f(l∗DG). But 2f(l∗DG) +

lf ′(l∗DG) ≥ 0 because of SOCDG, thus the slope is non-negative.

Chapter 2

Innovation and Patent Protection in

the Pharmaceutical Industry: a

Literature Review and Institutional

Details

2.1 Introduction

The pharmaceutical industry is noted for its investment in R&D and its reliance on

patent protection. Schweitzer (2007, pp. 22, Table 1.1) reports that the pharmaceutical

industry consistently has the highest ratio of R&D expenditure to sales among all indus-

tries. According to a report by PhRMA (2010) 1, in 2009 the industry in the US spent

an estimated $65.3 billion in R&D, or 16.0% of total sales. Cross-industry studies show

that patents are the most important in the pharmaceutical industry, relative to other

industries, in capturing and protecting the competitive advantages of new or improved

1Pharmaceutical Research and Manufacturers of America (PhRMA) is a trade group representingpharmaceutical research and biotechnology companies in the US.

33

Chapter 2. Pharmaceutical Innovation and Patent Protection 34

processes and products. (e.g. Levin et al. 1987, Cohen et al. 2002). Absent patent pro-

tection, reverse-engineering known drug compounds to produce a copycat is remarkably

cheap and easy, relative to the cost, time and risk associated with drug discovery.

In this chapter I start by describing the drug development process (Section 2.2) and

the phenomenon of follow-on drugs (Section 2.3) to establish the context for subsequent

analyses; in particular, I note the patterns of competition observed in the pharmaceu-

tical industry (Section 2.2.1). The above is necessary background information that the

analyses in Chapter 3 is based on: for example, the two-stage model in Chapter 3 are

based on facts on the drug development process and the timing of each stage; it also

utilizes assumptions about the costs which are justifiable in the industry context. Then

I review the following: (1) the literature on pharmaceutical entry (Section 4.1) and in-

novation (Section 4.2); (2) the theoretical literature (Section 5) related to the models

I develop in Chapter 3–this comes from two major sources, strategic entry deterrence

theories (Section 5.1) and optimal patents theories (Section 5.2).

2.2 The Drug Development Process

An understanding of the drug development process is important to capture the multi-

stage nature of decision-making in this industry. Such an understanding explains the

specification of the model in Chapter 3, and how the underlying assumptions follow

observables.

The process of drug development involves several stages. What follows is a modified

version of Schacter (2006)2. The focus is the US industry. The following flow chart

illustrates the critical decisions in the drug development process.

2For verbal descriptions also see DiMasi et al. (1991). IND stands for Investigational New DrugApplication; NDA stands for New Drug Application.


Research R

Preclinical Development D + F

Clinical trials : phases I, II, III C

Marketing

discovery

IND filing and approval

NDA filing and approval

Multiple decisions are involved in this process as follows:

Decision 1: should the company engage in research that may potentially lead to the

discovery of a pioneering drug?

Decision 2: if yes to decision 1 and the research is successful with the discovery of

a new drug, should the firm proceed with pre-clinical development, including laboratory

and animal testing?3 (In a separate process, patent filing to the PTO usually occurs

before or at the time of submission of new chemical entity (NCE) to the US Food and

Drug Administration (FDA).)

Decision 3: if yes to decision 2, should the firm carry out human clinical trials? The

clinical trials are in three phases, each in general more lengthy and expensive than the

previous phase. While these trials can take two to ten years, on average they take about

five years (Schacter 2006, pp. 8-10).

If the clinical trials are successful, the firm can file for a new drug application (NDA).

After approval it may start marketing the drug. The median time from NDA filing to

approval is 12.9 months (2005). About 20% of the drugs entering the clinical trial stage

3This depends on the perceived potential success and cost of the drug. And methods for producing asufficient amount of the drug with a consistent level of purity and potency has to be developed. The firmneeds to make plans, commit resources and organize the project for development. Before carrying outhuman trials, pharmacodynamic responses, metabolic profiling etc. in humans need to be establishedthrough the pre-clinical trials. See Schacter (2006, pp. 6 and 7) .


ultimately get approved (DiMasi et al. 2003).

Each stage of development is associated with risks. Despite the time and money

spent in the previous stages, a drug may not make it to the next stage. Reports on the

costs of R&D typically incorporate the costs of the drugs that are never marketed (e.g.

DiMasi et al. 1991, 2003). To reflect this risk, the model in Chapter 3 specifies four

different costs, where each includes a risk premium to reflect the risk of truncating the

development process.

Consistent with the existing literature (e.g. Grabowski & Vernon 1987), I recognize

that a pioneer firm requires an extra research phase that leads to the discovery of a first-

in-class drug, while its subsequent competitor does not. Consequently pioneer research

is costlier. For the pioneer firm, the assumption in Chapter 3 is that the research cost

R is incurred before the discovery of a pioneer drug. The competitor incurs a smaller

cost because the pioneering drug’s success has already indicated the fruitful direction

for research. Since for the purpose of Chapter 3, only the comparison between the two

firms’ costs matters, this subsequent research cost is normalized to zero. The preclinical

development cost is D for each firm plus a fixed cost F for the competitor. This devel-

opment cost D includes labor, raw materials, research costs of tests and estimates, costs

of designing production methods etc. The fixed cost covers equipments, initial training

etc. specific to the therapeutic class. For either firm, any fixed cost is incurred only once

for drugs in the same class. Thus, relative to the competitor, if the originator develops a

follow-on of its own pioneer drug it has the advantage of saving the fixed cost F . Clinical

trials cost C for each drug.4 These costs and their differences between incumbent and

entrant are important for the results that follow in Chapter 3.

4Marketing costs are of a significant proportion in the total cost of the industry; however, since it isnot relevant to this analysis, it is omitted in Chapter 3.


2.2.1 Competition in the Pharmaceutical Industry

After the pre-clinical trials, an IND (Investigational New Drug application) has to be

filed and approved by the FDA for the human clinical trials to begin. The FDA has 30

days to review an IND.

Different authors report different impacts (from slight to moderate) on existing drug

prices from the entry of a follow-on drug. For example, Lu & Comanor (1998) using

US data found that introducing an extra drug with “little or no therapeutic gain” (as

categorized by the FDA) typically brings on a reduction in price of 2%. In a Swedish

study, Ekelund & Persson (2003) find that a follow-on is actually twice the price of

existing therapies. A study discussed in DiMasi & Paquette (2004), on the other hand,

puts the average launch price discount at 14% for the new entrants relative to the mean

of existing drugs.

In contrast, generic entry against the original branded product induces significant

price competition: once the pioneer drugs patent expires, generics enter the market

and intense competition precipitates a significant prices reduction for all drugs in the

therapeutic class. Also, recent studies find that most of the market shares of top-selling

brand-name drugs are captured by generics within weeks of generic entry (Grabowski

2004).

Should the competitor realize positive profits from entry, the competitor will develop

a follow-on drug without intervention from the originator. The originator has the option

of developing a follow-on drug as a strategy to preserve its market share.

To develop a follow-on drug, the competitor has to pay a fixed cost (in terms of

equipments, initial training etc.) in addition to a development cost (in terms of labor

etc.). The originator has the advantage of not having to pay any fixed cost to develop

a follow-on drug, as any such cost is paid during the development of the original drug.

The originator still faces a a development cost.


2.3 The Controversy around Follow-on Drugs

The details of follow-on drug development are complex. Yet an understanding of their

development sequence is critical for any model of this market.

Follow-on drugs, pejoratively called “me-too” drugs, are a focus of controversy. Di-

Masi & Paquette (2004) define a follow-on drug as “a new drug entity with a similar

chemical structure or the same mechanism of action as that of a drug already on the

market. That is, a me-too drug is a new entrant to a therapeutic class that had already

been defined by a separate drug entity that was the first in the class to obtain regula-

tory approval for marketing.” In short, while follow-on drugs resemble pioneer drugs in

chemical structure and mechanism, they are distinct chemical entities and as such do not

infringe on the pioneer patent.

Approval of a follow-on drug does not require evidence of any advantage of the drug

over existing pioneer drugs. Consequently, while follow-on drugs are plentiful, many

have entered the market with limited clinical evidence as to their advantage over existing

pioneer drugs.

Critics argue that R&D expenditures on such drugs provide little or no added efficacy,

that there are too many of these drugs5, that the resources allocated to developing these

drugs should be reallocated to seeking more innovative therapies.6 Supporters point out

that follow-on drugs provide more therapeutic options, and since patients may respond

to similar drugs differently, and may also have different needs regarding dosing schedule

or delivery system, more choice is clearly a plus. What I do not do here or in Chapter

3 is evaluate the therapeutic pros and cons of follow-on drugs or the welfare effects of

5“From 1998 through 2003, 487 drugs were approved by the US Food and Drug Administration(FDA). Of those, 379 (78%) were classified by the agency as ‘appear[ing] to have therapeutic qualitiessimilar to those of one or more already marketed drugs,’ and 333 (68%) weren’t even new compounds(what the FDA calls ‘new molecular entities’), but instead were new formulations or combinations ofold ones. Only 67 (14%) of the 487 were actually new compounds considered likely to be improvementsover older drugs.”(Angell 2004)

6See, for example, Angell (2004), Hollis (2004), Avorn (2004), Goozner (2004).


follow-on drugs. My goals are different. In Chapter 3, my first objective is to analyze the

firm and product dynamics of decision-making around follow-on drug developments. My

second objective is to assesses the impact of different policy interventions (in particular

the increase of patent terms) on pioneering research.

2.4 Strategic Entry Deterrence

2.4.1 Strategic Entry Deterrence Theories

The models in Chapter 3 are related to existing literatures on strategic entry deterrence

and optimal patents.

The theoretical literature on “strategic investment to deter entry” is large. The strate-

gic instruments include limit pricing, excess capacity (Spence 1977, 1979, Bulow et al.

1985a), brand proliferation (Schmalensee 1978) and excess, unused patents (“sleeping

patents”) (Gilbert & Newbery 1982, Reinganum 1983)7. This paper is closely related to

the first of these, the use of excess capacity to deter entry.

Limit pricing is one instrument of entry deterrence(e.g. Bain 1949, Modigliani 1958,

Sylos-Labini 1962, Gaskins 1971), where the incumbent charges prices below the monopoly

price to make new entry appear unprofitable. For limit pricing to work, one assumption

is that the incumbent can commit to its pre-entry outputs despite entry. The limit out-

put is the minimum committed output that leaves insufficient residual demand for the

potential entrant; and the corresponding price, the limit price, is the maximum price at

which entry is deterred (Modigliani 1958). Yet whether the incumbent is able to commit

to the limit output or price is a problem: if the potential entrant is forward-looking and

rational, limit pricing may not be credible (e.g. Dixit 1979, Gilbert 1989) since it may

be in the incumbent’s best interest to raise the price post-entry, unless the incumbent

7See Tirole (1988) for an extensive overview of strategic investments to deter entry.


has private information about its costs and uses prices to signal low costs (Milgrom &

Roberts 1982). In the pharmaceutical market, however, such commitment may be pos-

sible because of price regulations in many pharmaceutical markets: once a price is set,

in general it can not be increased. This makes this market a good testing ground for

limit pricing. I will discuss the empirical findings of limit pricing in the pharmaceutical

industry in the next section, Section 2.4.2.

The traditional model of excess capacity building for entry deterrence has three steps 8:

(i) the incumbent makes an investment with the knowledge of post-entry equilibrium,

and the investment is sunk – it can not be retracted; (ii) the potential entrant observes

the investment and decides whether to enter; (iii) monopoly with no entry or duopoly

with entry profits are realized; the post-entry game is usually modeled as Cournot-Nash

competition. Dixit (1980) utilizes such a model in which the investment is production

capacity, and the pre-commitment of capacity lowers the ex post marginal cost of pro-

ducing up to the capacity. This is a response to Spence (1977), who also presents a model

where capacity lowers the marginal cost, yet Spence’s (1977) conclusions are based on

an imperfect equilibrium–the threat of the incumbent producing at a level equal to its

pre-entry capacity is not credible. Thus, while Spence concludes that firms may hold

idle capacity to deter entry, Dixit’s model with subgame perfect equilibrium predicts

that firms never hold idle capacity. Ware (1984) argues that if both the entrant and the

incumbent have to incur a sunk cost before incurring variable costs, then the entrant’s

decision of installing capacity should be modelled as a separate step, before the pro-

duction step takes place: if the incumbent’s installed capacity is a sunk cost, then the

entrant’s installed capacity is also sunk. This contrasts with Dixit (1980), where there is

no separate step of the entrant incurring the sunk cost of production capacity. In Ware

(1984), due to the ability of the entrant to commit by installing capacity, the strategic

8In sequential games, “steps” are usually called “stages” of the game; however, in this paper as I usethe term “stage” for something different, I will use “steps” to denote the sequence of decisions by therelevant players.


advantage of the first mover is lessened. Bulow et al. (1985a) point out that Dixit’s result

of no idle capacity critically depends on the products of the two firms being strategic

substitutes9. If the two products are strategic complements at a relevant range, then the

incumbent may hold idle capacity to deter entry: and if entry were to occur, it would be

optimal for the incumbent to use the excess capacity.

My result in Chapter 3 can be interpreted as the incumbent holding excess capacity

to deter entry: with some range of parameter values, the incumbent may produce a drug

(as in the first model) or develop a follow-on drug without either testing or marketing it

(as in the second model) simply to deter the entry of the competitor. This result doesn’t

depend on the complementarity of the two drugs, though the basic intuition is the same:

the incumbent will only invest in idle capacity (to produce or develop a follow-on drug

without marketing it) if it serves the purpose of deterring entry. If entry is certain, the

incumbent may not invest in a follow-on drug at all, depending on whether producing a

drug is profitable post-entry: if it is profitable, then the incumbent produces the follow-

on drug and markets it (the capacity is not idle in this case); if it is not profitable, then

the incumbent would not invest in it in the first place.

Eaton & Lipsey (1980) point out that the effectiveness of excess capacity depends on

the durability of capital. For example, capital that lasts forever and is sunk constitutes a

strong commitment on the part of the incumbent. The investment in follow-on drugs does

not last forever as generic competition eventually reduces profits; still, similar to capacity

building, patent protection offers entry deterrence for the period of the combined rights

in the patent. Also, for the potential entrants, drug development is a complex process

involving consecutive decisions. Thus in Chapter 3 the traditional structure of excess

capacity building with one stage of entry is modified to contain two stages of entry (in

the second of my two models) to fit industry facts: a firm not only decides whether to

9Product 2 is a strategic substitute for product 1 if ∂2π1/∂x1∂x2 < 0 and a strategic complementif ∂2π1/∂x1∂x2 < 0. Here x1 and x2 are outputs of product 1 and 2 respectively, and π1 is the profitearned by product 1. Marginal profit decreases if more strategic substitutes are produced.


develop a new follow-on drug initially, but after development, decides whether to carry

out clinical trials.

Gilbert & Newbery (1982) conclude that the incumbent firm will always engage in

preemptive patenting, on the basis of the claim that the monopoly incumbent is will-

ing to invest more for the new technology than the potential entrant. This requires the

competitive condition that, with an entrant and competition, total industry profits will

be less than the incumbents profit with both old and new technology. The condition

is compelling if both firms were to develop the same new technology. For the result of

Chapter 3, such a condition need not hold, where the incumbent engages in preemp-

tive patenting, and the incumbent and entrant contemplate the development of different

follow-on drugs. Further more, Gilbert & Newbery (1982) require a free-entry condition

for potential entrants. This has no role in the model developed here. The absence of free

entry accords with the realities of the pharmaceutical industry, where brand loyalty, sunk

costs, advertising by incumbents are all known to act as entry barriers. Also, Gilbert &

Newbery (1982) assumes that R&D decisions are once-and-for-all commitments made at

the start of the patent race. While this assumption is also used in the first of the two

models in this paper, the second model captures the fact that pharmaceutical R&D pro-

cess requires a series of decisions, which depend on the choices, both past and anticipated,

of rival firms.

Reinganum (1983) argues that the results of Gilbert & Newbery (1982) depend on

their assumed deterministic innovation process, and if the process is stochastic, the in-

cumbent firm may not engage in preemption: when the process is stochastic, the marginal

benefit of reducing investment in R&D is more for the incumbent, so it invests less than

the challenger; but when the process is deterministic, the incumbent would not invest

any bit less than the challenger, since investing infinitesimally less would result in a non-

infinitesimal loss in profits when the challenger succeeds in obtaining a new patent. Here

without assumptions of uncertainty, there is the possibility of no preemption. Since the


two firms potentially develop different products, if the incumbent is unable to deter the

challenger through new patents, it is better off not investing at all. This occurs when

the challenger stands to gain by investing even if the incumbent first develops a patented

new product.

Fudenberg et al. (1983) model a multi-stage patent race among identical firms and

indicate that the result of Gilbert & Newbery (1982) (the leader in the race is persistently

the incumbent firm) does not require a deterministic technology for R&D, as long as the

lead firm can ensure that it remains the favorite at every stage of the race. Fudenberg

et al. (1983) is a multi-stage patent race that allows leapfrogging and promotes competi-

tion. Leapfrogging occurs not only when the probability of success is a stochastic function

of R&D effort. It also may occur if there is an information lag as to the rival firm’s ac-

tivity: The R&D process is required to pass through a number of discrete steps, and in

each period the firms simultaneously decide how many steps to advance and advancing

more steps require more costly R&D effort. Information lags occur because each firm

must choose its R&D effort without knowing what rival firms are doing in the current

period. That gives a follower the possibility to catch up with the lead firm, provided it

expends more effort. Fudenberg et al show that a firm will continue to pursue a patent

as long as it is no more than one step behind the leader, in which case it has the hope of

leapfrogging, and will otherwise drop out of the competition.

2.4.2 Deterrence of Generic Entry

There is a vast empirical literature on the entry deterrence of generic drugs by brand

name firms. To my knowledge the literature has so far ignored the entry deterrence of

brand-name drugs by the producer of another brand-name drug.

Bergman & Rudholm (2003) study the pricing behavior of brand-name drug manufac-

turers in Swedish markets when facing potential (when patents expire but before generic

entry) and actual entry by generics. They find that prices decline in response to both


potential and actual entry, consistent with limit price theory. The effect of potential

competition from generics is significant; it is of the same magnitude as the effect of one

additional actual entrant. Similarly, Cool et al. (1999) study the U.S. pharmaceutical

industry during 1963-82 and conclude that potential competition reduces the profitabil-

ity of brand-name firms. Caves et al. (1991) examine the effects of patent expiration

and generic entry for brand-names that lost their patent protection between 1976 and

1987. They find the price of brand-names falls with generic entry, and the rate of decline

is small and diminishes with successive entrants. They also find that after the patent

expires, the price of brand-name drugs actually increases until generic entry, indicating

no evidence of limit pricing.

Aside from price, advertising is studied in the literature on the pharmaceutical indus-

try. Advertising serves as a barrier to entry if it generates scale economies by enhancing

demand that causes costs per dollar of revenues to decline (e.g. Spence 1980, Schmalensee

1983). Hurwitz & Caves (1988) find that advertising activities of brand-names preserve

their market shares against incursion by generic entrants. Their study uses simultaneous

equations to explain two endogenous variables: the market share held by the brand-name

drug (relative to its generics) and the ratio of its advertising outlay (flow expense) to its

sales revenue. In this case, the market share preserving effects are achieved by advertis-

ing outlays, independent of the “good will stocks” of a drug brand built up during the

patent protection period. Caves et al. (1991) postulates that advertising by brand-name

drugs right before or at the time of generic entry may exert a “signal-jamming” effect

on the promotions by generic entrants, hence advertising can serve the purpose of entry

deterrence particularly at these moments. The entry deterrence effect would thus im-

ply increased brand-name advertising in response to threatened or actual generic entry.

Their data includes thirty brand-name drugs spanning seven therapeutic classes. All

sample drugs faced generic competition, and data on these drugs include sales revenue,

proportion of sales through pharmacies, quantities sold, sales-promotion expenditures


and patent protection dates. Their study uses a nonstructural approach. They find that

advertising by brand-name firms decreases due to impending and actual generic entry,

contradicting the prediction of deterrence effect, instead suggesting that expanding the

overall market is the main effect of brand-name advertising. Grabowski & Vernon (1992)

analyze 18 brand-name drugs facing generic entry. They authors observe that the gener-

ics enter the market at significantly discounted prices relative to the brand-names and

continue to decrease the prices after entry. However, the prices of the brand-name drugs

actually increase in nominal terms shortly after generic entry and remain higher than

generics long after generic entry. As the market share obtained by generics increases over

time, the average market price of the drug decreases. The authors examine the effect of

generic entry on the pricing pattern of pioneers and conclude that there is no evidence

that the brand-name companies decreased their nominal prices in response to the much-

lower generic prices. They propose that such a phenomenon can be understood in terms

of a segmented-market model, where brand-name drugs keep a large brand-loyal mar-

ket segment despite the significant price disadvantage. With a reduced-form equation,

the authors also examine the factors influencing generic entry and prices. They explain

the number of generic entrants by existing price-cost markups of the branded products

(the percentage markup of brand-name price over generic marginal cost at the time of

entry, a measure of profitability of the entering generics) and brand loyalty entry barrier

variables, including number of years on patent (a proxy for the brand-name’s goodwill

stock), advertising of the brand (promotion to sales ratio of the brand-name in the year

before first generic entry). Unsurprisingly, they find that higher expected profits lead

to greater entry. Yet neither of the brand-loyalty variables in the above regression is

significant, indicating no evidence of the existence of entry barriers; in particular, they

conclude advertising does not serve as a means of entry deterrence from the incumbents.

As a response, Scott Morton (2000) notes that the advertising choice of a brand-name

drug is endogenous: for example, the price-cost markup, the advertising, and the number


of entrants can all be higher due to unobserved higher expectation of future profits. This

is to be discussed below.

In an empirical study Scott Morton (2000) evaluates the role of pre-expiration brand

advertising as a means of deterring generic entry into the US pharmaceutical market,

with revenue, quantity and advertising data from mid eighties to early nineties. In a

reduced form regression, she explains the amount of generic entry with the market size

of the brand-name drug (revenue is likely correlated with profits because marginal cots

are low in the pharmaceutical industry), the revenue share of the brand-name drug sold

to hospitals rather than drugstores (considering the possibility that hospitals are more

likely to buy generics), the number of substitutes of the drug that have already gone

off patent, advertising expenditure by brand-names starting three years before generic

entry (when generics are making decisions whether to enter) and a few dummy variables

specifying, for example, whether the drug treats a chronicle condition, and whether the

drug is injectable or topical (capital equipment and/or FDA requirements are potentially

different for these two drug forms). After instrumenting10 to correct the aforementioned

endogeneity problem present in the previous literature (e.g. Grabowski & Vernon 1992),

the coefficient signifying the effect of advertising on generic entry becomes insignificant.

She concludes that brand advertising is not a barrier to entry by generic firms. This result

is consistent with the traditional interpretation of advertising as a means of expanding

the market instead of defending the market share against therapeutic substitutes.

Ellison & Ellison (2007) evaluate the possibility of strategic entry deterrence of generic

drugs by brand-names. The deterrence strategies examined include advertising, product

proliferation and pricing. The theoretical part of their paper establishes that under cer-

tain conditions (to be explained), absent entry deterrence motives, the investment by the

brand-name firm is either increasing or decreasing throughout (ie. monotone increasing

10The instruments used by Scott Morton (2000) include the number of years on patent, whether thereexist other forms of the brand-name drug under patent protection, total firm promotion expenditures,and number of target physicians that could be expected to use the drug.


or decreasing) in market size (the number of potential consumers in the market): The

authors identify two potential effects of market size on the investment of brand-name

firm, “direct effect”11 and “competition effect”12. In the case both effects are positive,

the investment is monotonously increasing in the market size; in the case they are both

negative, the investment is monotonously decreasing in the market size; even if they are

of different signs, the authors argue the net effect is still likely monotone, as they believe

which effect dominates is unaffected by market size. The authors argue that entry de-

terrence motives undermine the monotonicity of investments in market size: the motives

are stronger in medium-sized markets than in small (no entry threat from competitors)

or large markets (entry from competitors can not be deterred). The empirical evidence in

their paper suggests non-monotonicity, evidence of entry deterrence behavior: Consider

investment in advertising by the incumbent as the strategic instrument available to deter

a potential entrant. The authors consider three types of investments that could have en-

try deterrence effects: advertising (average annual advertising three years before patent

expiration), presentation proliferation and pricing. Using a reduced form approach, the

authors look at investment of brand-name drugs both cross-sectionally and over time

(long before patent expiration vs. shortly before patent entry). They find that both

small and large markets have less advertising than intermediate markets; also, presenta-

tion proliferation increases shortly before patent expiration. In the subsequent Chapter

3, in the case of patenting, new innovations as strategic investment is is expected to ex-

hibit a non-monotonicity of a different kind: a pioneer firms invests in a follow-on drugs

in very large markets because of the great potential for profit; it does not invest in very

small markets because there is little potential for profit; it may or may not invest in

11The direct effect is positive if the marginal benefit of investment increases more than the marginalcost when increasing market size. With larger market size, a positive direct effect increases the incentiveto invest from the incumbent.

12The competition effect is positive if the marginal benefit of the investment is larger when theincumbent is in duopoly competition than when it is a monopolist. Since with larger market size, thecompetitor is more likely to enter, a positive competition effect increases the incentive to invest fromthe incumbent for larger markets.


intermediate markets, depending on the strategic entry deterrence motive.

2.5 Optimal Patent Length and Breadth

Aside from strategic entry deterrence, the models in Chapter 3 are also related to existing

literatures on optimal patents.

There is a vast literature on optimal patent length and breadth, starting with Nord-

haus (1969a,b, 1972), Scherer (1972). Most of these studies focus on the tradeoff between

static monopolistic inefficiency and the dynamic benefits of innovation. What is im-

plicitly assumed in these papers is that the longer the patent protection, the greater

the incentive to increase R&D expenditures. For example, under such an assumption,

Tandon (1982) examined compulsory licensing as a policy tool in response to monopoly

distortions generated by patents, with the conclusion that optimal patent has an infi-

nite life with an optimal royalty rate for a compulsory licensing scheme. He compared a

monopoly rent stream over a limited horizon to a smaller rent stream because of product

competition over a longer (infinite) horizon. Competition through compulsory licensing

leads to lower price-cost margins (with obvious consumer benefits). The two alternatives

are such that the present value of the rents to the inventing firm is the same under the

two alternatives so that the inventing firm is no worse off. Thus incentives for R&D

remain unchanged but consumers benefit through enhanced product competition. This

result is similar to Gilbert & Shapiro (1990), who analyse the tradeoff between patent

length and breadth13. They find that optimal patents call for infinite length with the

breadth adjusted to provide a pre-specified reward for patenting. In contrast, Klemperer

(1990) studied the tradeoff between the length and the scope of coverage14 of patents,

and concluded that either infinitely long, narrow patents or minimally short but very

13The “breadth” here means the flow rate of profits available to the patentee during the patent life14Here by scope of coverage Klemperer (1990) means the region of differentiated product space that

can be included in the patent.


broad patents are optimal. None of these examines the impact of patent length on the

incentive to innovate; instead, they take as given that longer patent length increases the

incentive to innovate and discuss when the static inefficiencies of monopoly is justified

by the benefits of such increases in incentives.

Gallini (1992) notes that long patents encourage imitation and may reduce the profit

of the original inventor. In her analysis, there is a given imitation cost for an innovation

that is not prohibitively high. When the patent life is longer, a rival is more likely to

imitate the patented product. The innovator has two choices after the innovation, either

patenting the innovation and risk imitation from rivals, or keeping it secret and risk

rivals learning about it anyway. When the patent length is above a certain threshold,

free entry of imitators dissipate the profits. This has feedback effects on the decision of

the innovator: in this case, the innovator prefers to keep the innovation secret instead of

patenting it. Since by foregoing patenting, the innovator also foregoes monopoly profits,

increased patent length does not necessarily encourage innovation because the innovation

may not be patented. Thus broad patents (patents that not only cover the original

inventions, but also inventions similar to them) that discourage imitators are optimal.

Aside from the decision whether to patent the innovation, the innovator in Gallini (1992)

doesn’t have an option to deter entry. In Chapter 3 the innovator may be able to deter

entry with its own secondary innovation, but there is no option of secrecy. Because of

the ease of reverse engineering, keeping the chemical structure secret is unrealistic in the

context of the pharmaceutical industry. The entry deterrence structure in my models

further develops the Gallini (1992) idea of increased patent life encourages imitation and

hence possibly reduces innovation: I also conclude that pioneer research is discouraged

relative to follow-on research; inexpensive R&D projects are discouraged, and ceteris

paribus expensive projects are favored instead, especially those with large clinical trial

costs. In Chapter 3, I focus only on the length of the patent because breadth of patents

is less flexible for the pharmaceutical industry–a different chemical structure obtains a


different patent, even if the physiological responses could be similar.

Weyl & Tirole (2010) argue that the above literature, using patent length and breadth

as screening variables for optimal patents, is severely restricted in its selection of instru-

ments. They develop a multi-dimensional screening model to address the optimal form

of reward for innovation. Their paper considers the trade-off between using prizes and

market-power (as conferred by intellectual property rights, including patents) as rewards,

quantifying the trade-off by choosing the price the social planner should induce innova-

tors to charge. Such a price is expressed as a fraction of the monopoly price: pure patents

would be represented by charging 100% of the monopoly price, while pure prizes would

be represented by charging zero. They develop tools to solve for the optimal degree of

market power and conclude that neither pure patents nor pure prizes are ever optimal.

2.5.1 The Effect of Patent Length on Innovation

This section explores the relation between the length of the patent in the pharmaceutical

industry and the incentive to innovate. Although counterintuitive, longer patents can

decrease the incentive to innovate for pioneer drugs.

When sunk development costs are included, industry folklore is that on the whole

the pharmaceutical industry does not make excessive profits. Associated with the sunk

development costs for pharmaceutical products is a quasi-rent stream. Underwriting

initial R&D requires assurance of the flow of these quasi-rents.15 As a result, it may seem

evident that a longer patent life and higher drug prices are stimuli for R&D spending and

consequently newer, better medicines16. These policy implications, however, are not this

straight-forward. What is missing is an account of the incentives to develop follow-on

drugs and the consequent interaction between pioneering pharmaceutical companies and

15See e.g. Scherer (2001).16In this paper I focus on patent life but not price. For apparently reasons, raising prices is highly

controversial and difficult to implement in reality, but increasing effective patent terms have occurredmultiple times in history and remain a policy possibility.


their follow-on rivals. When the incentives of potential rivals producing follow-on drugs

are taken into consideration, the pioneer firms may have incentives to deter the entry

of their competitors. The longer the patent term, the more incentives for the potential

rivals to produce follow-on drugs and more incentives for the pioneer firms to deter entry.

Consequently, the resulting net effect of longer patents is no longer straightforward and

will be explained in detail in Chapter 3.

Grabowski & Vernon (1987) develop a simulation model to examine the effects of

generic competition, regulatory review time and patent life on innovation levels and in-

dustry structure. At the start of simulation, they assume there are twenty equal sized

firms, each pursuing one of the two different types of R&D strategic behaviour, “pi-

oneering” and “imitative” R&D that competes with the pioneering firms in the same

therapeutic class. The former generates high levels of innovation, involving the develop-

ment of a new therapeutic class with the potential of achieving significant therapeutic

advances. The latter generates incremental innovations by investigating a known class of

drug products for developing marginal therapeutic advances. Pioneering R&D generates

higher revenue when successful, but is also more costly and takes longer. While post-entry

competition is not explicitly modelled, in terms of sales revenues, the pioneer product

introductions are more market expanding than redistributive, and the imitative product

introductions are largely redistributive. The authors assume that each firm pursues only

one pre-determined type of strategy and there is no entry of new firms, while exit occurs

after prolonged periods of low revenues. Also, each firm allocates a “target”percentage

of their net revenue into R&D, funding new projects provided that ongoing projects are

adequately funded first. Moreover, in their model, pioneers firms are unable to predict

the profitability of particular projects and unable to react to competition from both other

brand-names and generic producers. These simplifying assumptions are inconsistent with

expected profit-maximization behavior. In their simulation, Grabowski & Vernon found

that an increased rate of generic competition reduces both R&D level and net revenue


of pioneer firms, while an increase in effective patent life restores the level of innovation

and revenue. Not surprisingly, these findings follow directly from their assumptions. In

particular, the lack of endogenous investment decisions and the lack of strategic inter-

action between the two types of innovating firms are the key to the monotone relation

between patent life and innovation in their paper.

2.5.2 Contributions of Chapter 3

My contribution follows from the impact of capacity building on entry deterrence, which

has a long economic tradition. What’s new are (i) the association of entry deterrence with

an incentive to innovate; (ii) the application of this basic idea to competition between

brand-name drugs in the same therapeutic class whereas past contributions focus on the

interactions of brand-name drugs with their generic competitors; (iii) the recognition that

competition between brand-name producers is quite different from competition between a

brand-name producer and a generic producer. In general, entry deterrence adds another

important dimension to understanding the economics of the development and production

of follow-on drugs.

Chapter 3, to a lesser extent, is also related to the optimal patent literature. Instead

of discussing optimal patent length and breath in terms of social welfare, my contribution

is to focus on the anomaly of longer patents creating disincentives to innovate.

Chapter 3

Arrested Development–The

Unexpected Effect of

Pharmaceutical Patent Protection

on Innovation

3.1 Introduction

The descriptive evidence on the pharmaceutical industry presented below yields some

puzzling facts. Here are the facts outlined; the details and evidences are to be presented

later.

1. Industry observers claim that the new drug discovery “pipeline” is thinning and

new drugs are discovered at a lower rate than before1–a fact that threatens the

profitability, perhaps even the very existence of some pharmaceutical companies.

This remains one of the major concerns for pharmaceutical companies in recent

years. To mitigate the trend, pharmaceutical companies are expected to fill the

1See e.g. multiple articles from New York Times and Pharmacoeconomics.

53

Chapter 3. Arrested Development 54

gaps in their pipelines by various means, including closely following potential leads

to develop more products; but promising potential drugs continue to be cancelled

mid-development for non-medical reasons.

2. New technologies enhance efficiency in many areas of pharmaceutical R&D; but

R&D costs in the pharmaceutical industry have increased sharply.

3. The nominal optional length of marketing exclusivity granted to each drug has

increased through legislation changes over the years; but for new pioneer drugs

(first-in-class, as opposed to follow-on drugs) the average life of exclusivity in the

market (i.e. being the unique drug in the therapeutic class) has decreased.

What follows is an analytical explanation for these facts.

The pharmaceutical industry receives special treatment. The protection for innova-

tion constitutes not only regular patents (as in every other industry), but also exclusive

marketing rights. This exclusive privilege means that drugs with the same chemical

structure can not be sold before such marketing rights expire2. Policies on “patents”

in the pharmaceutical industry can be separated from general patent policies. For the

pharmaceutical industry, patents are entwined with exclusive marketing rights. Both

serve the purpose of keeping away copycat products. In this paper I use “patents” to

refer to both of these exclusive guarantees.

The following provides further descriptive evidence to support the puzzling facts out-

lined above.

2Patents in the pharmaceutical industry, including those of the substance, method of use, formulationand the production process, are granted in the US by the USPTO, like any other patents; however, theFDA may grant exclusive marketing rights to a brand-name drug independent of patents. Patents areusually applied for before clinical trials; exclusive marketing rights are granted when drugs are approvedfor marketing. During the time of marketing exclusivity of a brand-name drug, there can be no genericcompetition even if the relevant patents have expired.


In recent years, the claim is that the “pipeline” of new drug discoveries is thinning,

and innovation is in “crisis”3. New drug approvals per year have declined from 59 new

molecular entities (NMEs)4 in 1996 to around 20 per year in recent years5. What is

uncertain is whether this decline is transitory or permanent6.

Yet despite any concern over a thinning pipeline, the evidence suggests that new

drug development is often terminated for non-medical reasons. According to Walker’s

(2002) investigation of terminations of new product development among 28 pharmaceu-

tical manufacturers in 2000, 21.7% were based on “portfolio considerations” and another

16.2% were due to various factors other than clinical safety, toxicity and efficacy. If the

market supply for new and varied drugs were weakened, the expectation would be that

it is profitable to enhance (not reduce) drug development to market. This raises the

question: what could be the reasons for strategically terminating the testing of a new

drug despite any beneficial health potential for consumers? This paper addresses this

question.

Industry R&D spending has increased appreciably. According to the National Sci-

ence Foundation, R&D spending by global pharmaceutical and biotechnology industry

has grown more than six-fold over the past 25 years. For example, The Economist (June

16, 2005), citing CMR International, reports that global R&D spending was around $30

billion in 1994, rose to $54 billion in 2004 ($43 billion in 1994 dollars, a 43% increase), and

continues to rise. Such a rise in industry R&D spending, however, occurs simultaneously

with a thinning product pipeline. Together these suggest sharply falling research pro-

3See, for example, El Feki (2005) and The New York Times, Jan 11, 2006 “Drugs in ’05: Muchpromise, little payoff”.

4FDA uses NME to describe new compounds. The distinction between NME and later mentionednew chemical entity (NCE) is insignificant. NCE are new molecules or compounds that have not beentested in humans.

5The numbers are found on the FDA website. For example, from 2006 to 2009 the numbers are 19,16, 21, 18 respectively. Here the count aggregates “small molecule” New Drug Applications and “largemolecule” Biologics License Applications.

6See e.g. Cockburn (2006) Figure 1.2 or Cockburn (2004). Cockburn notes that the count of NMEsis a noisy measure and short-term fluctuations can be misleading. However, at the time this paper iswritten, the lowered NME count still persists.


ductivity7. Why aren’t there enough NMEs given such an increase in R&D expenditure?

This paper suggests one possible reason.

This decline in productivity occurs despite efforts to stimulate innovation by granting

more patent protection. For example, the US Hatch-Waxman Act of 1984 allowed patent

holders to extend the life of their patents to up to five years, compensating for delays in

clinical testing and FDA approval8; also, in 1995 the World Trade Organization treaty

increased the maximum patent life from 17 to 20 years9. Despite such increases in patent

life, DiMasi & Paquette (2004) report that the period of exclusivity in the market enjoyed

by a pioneer drug has fallen from a median of 10.2 years in the 1970s to 1.2 years for

the late 1990s. Why would an increased patent life coincide with a decreased exclusivity

period for pioneer drugs? This paper also addresses this question.

This paper provides a model of entry deterrence (with a few variations) that explains

all three puzzles in one unified structure. Moreover, other predictions of the model are

supported by observations in the industry: For example, the model predicts that a pioneer

firm sometimes produces its own follow-on drug before the patent on its successful pioneer

drug expires; the model predicts that clinical trial costs rise as a component of total costs.

The models yield counterintuitive policy predictions: for example, longer patent life may

distort incentives, inhibit innovation and result in lowered R&D productivity, consistent

with observations on the industry.

The theoretical construct in this paper is related to the standard entry deterrence

with excess capacity literature (e.g. Spence 1977, 1979, Dixit 1980, Bulow et al. 1985b)

reviewed in Chapter 2. In general the R&D models in this paper can be interpreted

7See e.g. Comanor (2007, pp. 67)8The Act also contains other provisions that in practice may extend patent life: for example, if a

brand-name company sues a generic company for patent infringement, FDA will delay approval of thegeneric drug for thirty months, in effect adding that to the exclusivity period of the brand-name drug.Similarly, by FDA Modernization Act (FDAMA) of 1997, if a drug is tested in children, it obtains sixmonths of extra protection, which also in effect adds to patent life.

9The actual increase is somewhat less than three years because the delay in approval now erodesinto the patent term.


as the incumbent pharmaceutical company holding excess capacity through product de-

velopment to deter entry. To capture the multi-stage nature of the pharmaceutical RD

process (development and testing), my second model has two stages: a firm not only

decides whether to develop a new follow-on drug (decision of the first stage), but after

development, decides whether to carry out clinical trials (decision of the second stage).

The organization of the paper is as follows: I start with a one-stage entry model.

Here “stage” refers to a stage of decision. In the one-stage model, firms only have one

stage of decision, in which they sequentially decide whether to produce a follow-on drug

to an existing first-in-class pioneer drug. Despite its simplifications, this one-stage model

serves two purposes: (i) because of its simplicity, common features of all of the models are

readily seen; (ii) the intuition that higher potential profit does not translate into higher

actual profit for the pioneering drug is also easily seen. Then the model is extended to

two stages, in which the firms also move sequentially. The difference is that there are

two sequential stages of decision, as outlined above.

Next I present a two-stage model which explains the three puzzling observations men-

tioned at the beginning of the paper. The model yields two possible outcomes: (i) under

some conditions, the pioneering firm’s ability to sink resources early as well as its ability

to cease further product development can permit this firm to deter the entrant from

profitably initiating follow-on drug development (hence the first observation); (ii) under

other conditions, when entry is not blockaded, the competitor enters the market, some-

times despite being unable to sustain head-to-head competition if it were to occur. The

“disincentive effect” of longer patents occurs when longer patents shift the equilibrium

from one in which entry is blockaded to one in which it is not: a potential project of pio-

neer drug research may become unprofitable as patent terms increase, since the resulting

failure in entry deterrence significantly reduces profits. I conclude that the disincentive

effect of longer patents exists under very general conditions, and such an effect is the

source of the distortion of incentives and lowered innovation productivity: longer patents


discourage inexpensive R&D projects, and expensive ones are favored for their effective-

ness in deterring entry; moreover, the costs of the chosen R&D projects increase despite

technological advances that reduce costs in general (hence the second observation); in

particular, projects that involve higher clinical trial costs are favored, resulting in an

increase in clinical trial costs relative to other R&D costs; pioneer research is discour-

aged as follow-on drug research is favored over pioneering research, and this decreases

the amount of time a pioneer drug is unique in its therapeutic class (hence the third

observation).

I also discuss a multi-period model with explicit timing and conclude that major

prediction of the model remain intact in such a setting.

Finally, I discuss other implications of the model and propose multiple further em-

pirical tests for the model.

3.2 The One-Stage Entry Game

3.2.1 General Setting

Two firms each decide whether to develop and produce a follow-on drug to an existing

first-in-class drug (or, “pioneering drug”) that was invented by one of the firms, firm 1

(also called the originator). Firm 2 is called the competitor. Assume each firm can at

most develop one follow-on drug. Different from most of the existing literature on pre-

emptive patenting since Gilbert & Newbery (1982), and more in line with the facts in the

pharmaceutical industry, I assume the firms’ potential follow-on drugs, while substitutes,

are sufficiently different from each other and will hold different patents once developed.

The timing is as follows: At the start of the game, there is already one drug, the

pioneer drug, discovered by firm 1. Also, the pioneer drug holds a patent, which by as-

sumption does not expire before the follow-on drugs enter the market if development is set

in motion immediately. And a follow-on cannot enter the market before the completion


of both development and clinical trial processes.10

The structure of the one-stage game is similar to an entry game. Firms sequentially

decide whether to enter the follow-on market, and if a firm enters, it automatically

continues with clinical trials once the drug is developed.

Full information is assumed. Each firm is able to observe all previous moves. Firm 1

moves first, deciding whether to enter the follow-on market (producing a follow-on drug).

In the following diagram, ‘Y ’ denotes “yes, enter” and ‘N ’ denotes“No, do not enter”;

πiab (i = 1, 2; a, b = Y,N) is the payoff to firm i when firm 1’s and 2’s choices are a and

b respectively. There is no discounting. Payoffs are explained below.

Firm 1

Firm 2

π1NN , π2NN

N

π1NY , π2NYYN

Firm 2

π1Y N , π2Y N

N

π1Y Y , π2Y YY

Y

3.2.2 Payoffs

A pioneer firm may develop a follow-on drug early for deterrence effects, or develop

late to replace the soon-to-expire patent of the first-in-class drug. Since strategic entry

deterrence is the effect I intend to explore, I focus only on early drug development 11.

This means that firm 1 chooses either to develop a follow-on drug early or not at all,

and when firm 1 develops the follow-on drug early, then for at least a while the follow-on

10In the one-stage entry model and the two-stage model, there are no exact timings, as no explicitassumptions about multiple periods are in the model settings; in the multi-period model to be discussedin detail in later sections, the timing of the start of the game can be varied, as long as it is after theoriginal patent and sufficiently before its expiry.

11This certainly doesn’t rule out developing another follow-on drug late, which is a different strategicconsideration from the focus of this paper.


drug would compete in the same market with the pioneer drug.

Even though a follow-on drug is sufficiently different from the pioneer drug to obtain

a separate patent, there is typically no evidence of a large market expansionary effect

in introducing a follow-on drug 12. So here I assume introducing a follow-on drug will

not expand the market, i.e. the total industry quasi-rent in the same therapeutic class

is non-increasing with the introduction of more brand-name drugs:

Assumption (A1). (No market expansionary effect) The total industry quasi-rent of a

therapeutic class is non-increasing in the number of brand-name drugs that exist in the

same class.

The market size, or total industry quasi-rent, is a parameter closely related to the

incentives for drug development of pharmaceutical companies. The market size reflects

both the population affected by a particular disease and its willingness and capability

to pay. This in turn is related to income levels of the affected persons. For example,

for any disease, Lichtenberg (2005) finds that the resources allocated to pharmaceutical

innovation is positively related to the burden of disease13 in developed countries (but not

to the burden of disease in developing countries).

I also use the following assumptions:

Assumption (A2). (Non-triviality, or, competitor is viable) If firm 1 doesn’t produce

a follow-on, it is profitable for firm 2 to produce a follow-on.

This assumption simply states that firm 2 has the potential to compete in the same

market, thus makes entry deterrence a non-trivial consideration for firm 1. In this case,

Firm 2 is called viable in the follow-on market.

Assumption (A3). (Common costs) The firms have the same development and clinical

12In fact, most follow-on drugs are very similar to pioneer drugs in terms of efficacy and side effects.See eg. Angell (2004).

13Here for empirical analysis of “burden of disease”, the author used the number of disability-adjustedlife-years attributable to a disease and in the case of cancer, the number of people diagnosed with aparticular form of cancer.


trial costs for drugs in the same therapeutic class, although the pioneer firm would

incur an extra cost for research into this new therapeutic class; also, the fixed costs

in developing a drug need be incurred only once for any single therapeutic class. The

potential heterogeneity of firms is partially captured in the research cost and fixed cost.

The costs are R (research cost), F (fixed cost), D (development cost) and C (cost of

clinical trials). Only the pioneer firm incurs R for the discovery of the pioneer drug in

the therapeutic class. Each firm incurs F only once for each therapeutic class. For each

drug, development costs D and clinical trials cost C.

With A1, A2 and A3, we can conclude that firm 1 must find itself better off from

deterring firm 2’s follow-on drug (incurring investment costs that lead to the production

of a follow-on drug), than not deterring (yet without incurring costs in developing a

follow-on drug either). The intuition is that if firm 2 finds it worthwhile to engage

in follow-on drug production without expanding the market, then firm 1 also finds it

worthwhile to deter firm 2 by incurring the same costs. Note that the aforementioned

assumptions are sufficient yet not necessary for such a preference of firm 1: in particular,

if the introduction of follow-on drugs do have moderate market-expansionary effects, or

if the costs of the two firms at each stage are not exactly the same, firm 1 may still

have such a preference. This shall be discussed in more details after I explain the payoff

rankings below.

The above means that firm 1’s payoffs can be ranked as follows:

Firm 1 I. Best: neither firm enters;

II. Second best: firm 1 alone enters;

III. Third and fourth best (unranked until further parameter values are known):

Firm 2 alone enters/both enter. If both firms enter, the statement is that the two firms

compete head-to-head; if firm 1 prefers competing head-to-head to letting firm 2 alone

enter, the statement is that firm 1 is sustainable in the whole game. Formal definitions


of sustainability for each firm will be in the detailed analysis that follows.

We also know firm 2’s payoffs can be ranked as follows:

Firm 2 I. Best: (positive payoff) Firm 2 alone enters;

II. (Unranked until further parameter values are known) (i)firm 2 doesn’t enter, re-

gardless whether firm 1 does; (ii)both firms enter.

We know that in case (i) firm 2 obtains zero payoff; in case (ii) firm 2’s payoff can be

positive, negative or zero. Firm 2 is sustainable in the whole game if this payoff is positive,

and non-sustainable in the whole game if it is negative. When firm 2 is sustainable in

the whole game, head-to-head competition is better for firm 2 than the outcome where

it does not enter.

Since no market expansionary effect is assumed, even if firm 2’s payoff is positive

when both enter (ie. firm 2 is sustainable in the whole game), the outcome in which both

enter is still worse for firm 2 than the case where firm 2 alone enters.

To discuss the payoffs in detail, I use further simplifying assumptions that are suffi-

cient but not necessary to the conclusions of this paper. As long as the payoff rankings

listed above remain, the best-responses and the equilibria are the same. This is so be-

cause the equilibria depend only on the payoff ranking instead of the absolute values of

the payoffs, and the predictions of the models remain since they are inferred from the

equilibria. Thus such assumptions serve only the purpose of reducing clutter in reasoning

and do not drive the results. For example, I will use the proof of C1, a corollary to be

explained later in this section, to show that payoff rankings for firm 1 may remain even

without simplifying assumptions such as A1 or the following A4.

Instead of A1 (market doesn’t expand), I further state the simplifying assumption

that market size is constant regardless of the number of brand-name drugs:

Assumption (A4)(A1’). (Constant market size) The total industry quasi-rent of a

therapeutic class remains constant regardless of the number of brand-name drugs that


exist in the same class.

A1 here indicates that firms are dividing a “quasi-rent pie” 14 independent of the

number of firms. Though A4 is a simplifying assumption, it enjoys empirical support:

Generally, if firms in an industry produce highly substitutable products, the sum of

industry profit decreases with the number of entrants. Follow-on drugs are not identical

as some product differentiation exists. Consequently, introducing more brand-name drugs

in the same therapeutic class can have a slight to moderate effect on reducing price 15.

What has also been claimed is that follow-ons can expand therapeutic options and expand

the patient base for the general class of drugs. This claim has weaker support from

market observations. These two effects have an opposite (possible off-setting) impact

on the market size and are both extremely moderate, offering support for a constant

“quasi-rent pie”.

Also, I assume that the total industry quasi-rent, unsurprisingly, increases with the

patent length. (The industry quasi-rent pie size P (L) increases continuously with patent

length L.) Time periods are not explicitly modelled here. One further simplification is

to hold constant the yearly quasi-rent “pie”: then it renders a per period return p for

all brand-name drugs and the total number of years (the length of the patent) is L, and

the total quasi-rent for the industry throughout the life of the original patent (assuming

generics enter and the quasi-rents disappear) is P and P = pL. But such simplification

is certainly unnecessary for my conclusions.

With assumptions A4 (i.e. A1’) (constant market), A2(non-triviality) and A3(common

costs), market shares are represented in a reduced form structure as follows:

14As usual, here the quasi-rents are calculated without considering the R&D costs R,F ,D and C, tobe explained later.

15 Lu & Comanor (1998) using US data found that introducing an extra drug with “little or notherapeutic gain” (as categorized by the FDA) typically brings on a reduction in price of 2%. And ina Swedish study, Ekelund & Persson (2003) find that a follow-on is actually twice the price of existingtherapies. A study discussed in DiMasi & Paquette (2004), on the other hand, puts the average launchprice discount at 14% for the new entrants relative to the mean of existing drugs.


1. If none of the firms enters the follow-on market, then the originator captures the

whole market in the therapeutic class. Sunk costs are R, F , D and C. In the

marketing stage, firm 1 obtains a return (quasi-rent) of P for the whole market.

2. If firm 2 is the only entrant to the follow-on market, then firm 2 realizes a market

share β (0 < β < 1), and firm 1 obtains the rest, 1 − β of the market.

3. If firm 1 is the only entrant to the follow-on market, then it still captures the whole

market. The returns continue to be P but there is an additional cost of D + C

incurred by firm 1 in developing the follow-on drug.

4. If both firms enter the follow-on market, firm 2 captures a market share α (0 <

α < β), and firm 1 captures 1 − α.

With the above payoff structure, A2 (non-triviality) states that (N, Y ) �2 (N,N) or

βP −F −D−C > 0. This means that firm 2 is viable if and only if βP −F −D−C > 0.

The following table summarizes the payoffs in each of the four scenarios. By the time

the game starts, the costs that firm 1 incurred to develop the pioneer drug are sunk

and therefore are ignored. The costs that the firms incur to develop the follow-ons are

included.

Firm 2

Firm 1Payoffs Y N

Y (1 − α)P − D − C, αP − F − D − C P − D − C, 0N (1 − β)P , βP − F − D − C P , 0

Table 3.1: Payoffs: one-stage game

Ties are avoided by assuming away any case where payoffs are equal in any two

scenarios.

Corollary (C1). By A4, A2 and A3, firm 1 prefers to be the sole producer of a follow-on

drug, instead of having firm 2 do so, ie. (Y,N) �1 (N, Y ).


Proof. With A4 and A3, A2 is equivalent to βP − F − D − C > 0. βP − F − D − C >

0 ⇒ βP − D − C > 0 ⇔ P − D − C > (1 − β)P ⇔ (Y,N) �1 (N, Y ).

As mentioned before, firm 1 can have the same preference (payoff ranking) as in C1

without satisfying A4, A2 and A3: as can be seen from the proof, these are sufficient but

unnecessary conditions.

To further rank payoffs of both firms by parameter values, I define the concept of

sustainability in the whole game in the context of the above payoff structure. Each

firm’s sustainability depends completely on parameter values that are exogenous.

Definition 1. Firm 1 is sustainable in the whole game if as long as firm 2 produces

a follow-on drug, firm 1 is better off also producing a follow-on drug than not producing

(ie. firm 1 being sustainable in the whole game is defined by (Y, Y ) �1 (N, Y )). Firm 1

is unsustainable in the whole game if (Y, Y ) ≺1 (N, Y ).

Being sustainable for firm 1 means that (1 −α)P −D−C > (1−β)P , or (β−α)P >

D + C; firm 1 is non-sustainable in the whole game if (β − α)P < D + C. The cases in

which (β − α)P = D + C is ignored.

Whether firm 1 is sustainable in the whole game depends on

• P : the size of the returns “pie” or total quasi-rents for the therapeutic class;

• 1−α and 1−β: how much of the market share firm 1 retains if firm 2 alone enters

and both firms enter;

• D and C: the scales of the pre-clinical development cost and clinical trial cost.

The greater P or β −α and the smaller D or C, the more likely firm 1 is sustainable.

Firm 2’s sustainability in the whole game is similarly defined:

Definition 2. Firm 2 is sustainable in the whole game if as long as firm 1 produces

a follow-on drug, firm 2 is better off also producing a follow-on drug than not producing


(ie. firm 2 being sustainable in the whole game is defined by (Y, Y ) �2 (Y,N)). Firm 2

is unsustainable in the whole game if (Y, Y ) ≺2 (Y,N).

Firm 2 is sustainable in the whole game if αP −F −D−C > 0; it is non-sustainable

in the whole game if αP − F − D − C < 0. The case in which αP − F − D − C = 0 is

ignored.

Whether firm 2 is sustainable in the whole game depends on

• P : the size of the “pie”;

• α: how much of the market share firm 2 can get if both firms enter;

• F , D and C: the scales of the costs.

The greater P or α and the smaller D or C or F , the more likely firm 2 is sustainable.

The diagram of the game with payoffs follows.

Firm 1

Firm 2

P , 0

N

(1 − β)P , βP − F − D − CYN

Firm 2

P − D − C, 0

N

(1 − α)P − D − C, αP − F − D − CY

Y

The subgame perfect equilibria (SPEs) in this game depends on whether the firms

are sustainable. Details follow.

3.2.3 SPEs of the One-Stage Game

Proposition 3. In the one-stage game, whether firms produce a follow-on drug is solely

determined by the sustainability of both firms in the whole game, as subgame perfect

equilibrium (SPE) outcomes are indicated in the following table:


SPE outcomes Firm 2 sustainable Firm 2 unsustainable

Firm 1 sustainable (Y,Y) (Y,N)

Firm 1 unsustainable (N,Y) (Y,N)

Because of the simplicity of the proposition, the formal proof is omitted.

Notice that I have excluded the possibility that firm 2 is unviable. But in this context,

it is straightforward to infer the outcome if firm 2 is unviable: If firm 2 is unviable, then

it has no incentive to enter the market to compete with firm 1. With the assumption of

constant market size, firm 1 does not have any incentive to produce a follow-on drug at

all if it were not for the purpose of entry deterrence. Hence, as long as firm 2 is unviable,

the outcome would be (N,N).

The following diagram summarizes the SPE outcomes. The directions of the arrows

indicate the directions where sustainability increases.

1’s sustainability


(Y, Y ) ∼1 (N, Y )

(Y, Y ) ∼2 (Y,N)

(Y,Y)

(Y,N)

(N,Y)

Figure 3.1: SPE outcomes depend on the sustainability in the whole game. The twocrossing lines (dashed vertical line, solid horizontal line) are where firm 1 and 2 each isindifferent between joining a head-to-head competition and staying out; they divide thesustainability into four regions. Each color corresponds to an equilibrium outcome. Forexample, the two purple regions correspond to the outcome (Y,N). For a “disincentiveeffect” (explained later) to occur, the equilibrium has to shift cross the solid line.


This yields the following interpretations:

Corollary (C2). When both firms are sustainable, they each choose to produce a follow-

on drug.

Corollary (C3). If only firm 2 is sustainable, it chooses to produce while firm 1 stays

out.

This is because firm 1’s entry would fail to deter firm 2, and absent deterrence, firm

1 receives no benefit from producing a follow-on.

Corollary (C4). Firm 1 produces a follow-on that deters firm 2 if and only if firm 2 is

unsustainable.

An interesting point (common to this and subsequent models) is that increasing the

size of the “quasi-rent pie” doesn’t necessarily benefit firm 1, and furthermore from a

pre-innovation perspective, firm 1’s incentive to develop the pioneer drug in the first

place may be reduced. For example, if for returns P1, firm 1 is the only firm sustainable,

and if for P1 + ε, both firms are sustainable, then if ε is small enough, firm 1’s profit

decreases from P −D−C to (1−α)(P + ε)−D−C. The reason is that greater returns

induce fiercer competition, and any incremental gain to returns may be insufficient to

compensate for a smaller share of those returns. The next section develops this further.

3.2.4 (Dis)incentive Effect of Longer Patents

What happens to the equilibrium if there is a shift in patent length?

Look first at two extreme cases: one in which the length of the patent L is very small,

the other very large.

In one extreme, consider the length of the patent just sufficient to keep firm 2 viable

(recall that firm 2 is viable if and only if βP (L) − F − D − C > 0): define Lmin as

βLminp−F −D−C = 016. If the patent length is Lmin, we know firm 2 is not sustainable

16Here for simplicity I assume P (L) takes linear form.


when both enter. This is because for firm 2 to be sustainable with both firms in the

market, π2Y Y (Lmin) = αLminp − F − D − C > 0 has to hold, but it doesn’t since

π2NY (Lmin) = βLminp − F − D − C = 0 and α < β. The equilibrium is (Y,N) in this

case. In the other extreme, L is sufficiently large that both firms are sustainable and the

equilibrium is (Y,Y).

Between the two extremes, the equilibrium may be (N,Y) at some L. Details are in

the Appendix.

It turns out that a small increase in L that shifts equilibrium outcome from (Y,N)

to (Y,Y)17 induces lower returns for firm 118, despite the increase in the “pie” size P (L).

I refer to this effect as the disincentive effect of longer patent length. The disincentive

effect occurs because such an increase in L has two effects: (i) a continuous increase

in “pie” size P (L) with the continuous increase in L; (ii) a “lumpy” decrease in Firm

1’s equilibrium share of P from 1 to 1 − α, and the latter outweighs the former. This

means that with the increase of patent life, there is both an increase of the total market

size (longer protection against generics) and a decrease of firm 1’s market share (more

competition from potential entrants). These have opposite effects on the returns to firm

1. If the latter effect outweighs the former, as is the case when a small increase in L just

changes the equilibrium from (Y,N) to (N,Y), the composite effect is a decrease to the

returns of firm 1.

I formally state the analysis in the Appendix in the form of the next proposition.

Notice that αLap − F − D − C < 0 means firm 2 is non-sustainable at La, thus the

equilibrium is (Y,N)):

Proposition 4. Given a patent length La and costs F , D and C that satisfy αLap −

F −D−C < 0, a longer patent life Lb (Lb > La) induces disincentive effect by switching

17In fact, as can be seen from the diagram in the Appendix, a shift in equilibrium from (Y,N) to(N,Y) also has a “disincentive effect”.

18If L is very large, say approaching infinity, obviously the returns for firm 1 is also going to be verylarge.


equilibrium from (Y,N) to (Y,Y) if and only if all of the following conditions hold:

(i) αLbp − F − D − C > 0 (firm 2 is sustainable at Lb);

(ii) (β − α)Lbp > D + C (firm 1 is sustainable at Lb);

(iii) Lb < La/(1 − α) (returns to firm 1 are higher with La).

The three conditions indicate that to achieve a disincentive effect for firm 1, Lb needs

to be long enough to change the equilibrium outcome (as indicated by the first two

conditions) but not too long (as indicated by the third condition).

An analogy can be drawn in the case of equilibrium switching from (Y,N) to (N,Y),

bringing about the disincentive effect.

Such an Lb doesn’t necessarily exist. I also state the analysis in the Appendix formally:

Proposition 5. Given La, F , D and C that satisfy αLap−F−D−C < 0, a longer patent

length Lb that induces disincentive effect (by switching equilibrium from (Y,N) to (Y,Y))

exists if and only if both αLap/(1−α)−F −D−C > 0 and (β−α)Lap/(1−α) > D+C.

These two inequalities mean that (i) initial patent life for comparison, La, needs to

be large enough; (ii) both firms need to gain a large enough market share by engaging in

head-to-head competition in the follow-on market: for firm 2, the gain in share is α and

for firm 1, β − α; both α and β − α need to be large enough. In other words, the initial

patent life for comparison, La, needs to be large enough relative to sunk costs adjusted

by market shares.

Recall that switching equilibrium from (Y,N) to (Y,Y) is only one of the two scenarios

that cause disincentive effect (the other one is switching equilibrium from (Y,N) to (N,Y)).

Thus the general disincentive effect exists under more general conditions. In the two-stage

model, I will discuss the existence of the disincentive effect in more detail.

Finally, this one-stage model does not answer why some drug development processes

are strategically terminated, as I asked in the introduction section. Notice that firm l

does not benefit from producing a follow-on in isolation: the market is shared with its


own pioneer drug, so that while producing a follow-on brings no extra profit, it does

bring extra costs at the development stages. The prediction of the current model is that

firm 1 fully produces a follow-on to deter firm 2. This result follows from the simplifying

but perhaps unrealistic assumption that each firm makes only one decision regarding

development. The next section considers the impact of relaxing this assumption in the

two-stage model.

3.3 The Two-Stage Model

In the drug development process, instead of one decision to develop a new drug, there

are in fact a series of decisions made at various stages. For example, even if a drug has

gone through pre-clinical development, the firm may decide against continuing with the

human clinical trial. To capture this sequential and discrete decision making in the drug

development process, I model the process as two sequential stages of decision making for

each firm that develops a drug19: (i) whether to enter the follow-on market by starting

to develop a drug (action “E” for entering, and “N” for not entering); and (ii) if the drug

is developed, whether to continue for clinical trial (action “C” for continuing, and “S”

for stopping).

If a firm chooses to develop a follow-on, we say it enters the follow-on market (action

E, which may or may not be followed by C); if a firm does not develop a follow-on, we

say it stays out (action N); if a firm chooses to both develop a follow-on and to continue

with its clinical trial (action E followed by C), we say the firm produces a follow-on.

Obviously, if a firm stays out, it can not produce a follow-on; but if it enters, it may not

produce a follow-on, depending on its decision with respect to the clinical trial.

Despite the diagram that follows, the description of the two stage game is simpler:

the firms take turns to move and full information is assumed. Once a decision is made,

19“It is customary to characterize new drug discovery and development in terms of time phases. Theprincipal dichotomy is between the pre-clinical and clinical phases.” (Scherer 2007)


costs are paid and no reneging can occur. Here is the time line:

Stage 1: Firm 1 moves first, deciding whether to develop a follow-on. Then, observing

1’s decision, firm 2 also decides whether to develop a follow-on.

Afterwards, development takes place if decided, and follow-on drugs are discovered.

Those events are observed by both firms.

Stage 2: Firm 1 moves first deciding whether to carry out the clinical trial if develop-

ment was decided in the previous stage; skip firm 1 otherwise. After observing firm 1’s

action, firm 2 decides whether to do clinical trial, if development was decided in the first

stage.

Note that if a firm chooses both E and then C, it is equivalent to choosing Y in the

one-stage game.

Firm 1: E/N

Firm 2: E/N Game Ends

Firm 1: C/S Game Ends

Firm 2: C/S Game Ends

if both choose N

if 2 chooses Nif 1 chooses N, 2 chooses E

if 1 chooses E

if 2 chooses E

3.3.1 Payoffs

I assume the costs associated to each stage is incurred at the beginning of the stage. This

is to capture the fact that much of the cost is sunk during the drug development process


regardless of success or failure.

The market share scenarios are the same as in the previous one-stage model, thus the

payoffs are also nearly the same, aside from the caveat that if a firm chooses E and S, it

still has to pay the development cost D and possibly F if it is the competitor.

The payoffs table follows.

Firm 2Payoffs E,C E,S N

E,C (1 − α)P − D − C, αP − F − D − C P − D − C, −F − D P − D − C, 0E,S (1 − β)P − D, βP − F − D − C P − D, −F − D P − D, 0N (1 − β)P , βP − F − D − C P , −F − D P , 0

Table 3.2: Payoffs in the whole game: two-stage model

Assumption A2 (non-triviality) still holds.

It turns out that the subgame perfect equilibria in this model critically depend on

whether each firm is sustainable, both in the whole game and in the second stage. Sus-

tainability in the whole game is already defined in Section 3.2.2; here I give definitions

for sustainability in the second stage.

Definition 3. Firm 1 is sustainable in the second stage if (C,C) �1 (S,C) in the case

that both firms have entered during the first stage, ie. (β−α)P > C; it is non-sustainable

in the second stage if (C,C) ≺1 (S,C) or (β − α)P < C.

Definition 4. Firm 2 is sustainable in the second stage if (C,C) �2 (C, S) in the case

that both firms have entered during the first stage, ie. αP − C > 0; it is non-sustainable

in the second stage if (C,C) ≺2 (C, S) ie. αP − C < 0.

We ignore all cases in which any firm is neither sustainable nor non-sustainable.

Remark 1. If any firm is sustainable in the whole game, then it must be sustainable in

the second stage; yet conversely, if any firm is sustainable in the second stage, it is not

necessarily sustainable in the whole game.

This can be easily seen from the definitions.


3.3.2 SPEs of the Two-Stage Model

I leave the technical analysis of the model to the Appendix and only present the results

here.

As the following proposition indicates, if both firms choose to enter, in the second

stage, any firm that is sustainable will surely go ahead and continue. And those who are

not sustainable have to “pause and think” about their opponents before they continue.

Proposition 6. If both firms have chosen to develop the follow-on drug in the first stage,

then in the second stage:

I. If both firms are sustainable in the second stage, the SPE outcome is (C,C);

II. If only firm 2 is sustainable in the second stage, the SPE outcome is (S,C);

III. If firm 2 is non-sustainable in the second stage, the SPE outcome is (C,S).

The proposition is summarized in the following table:

Firm 2

Firm 1Outcomes (C,C) �2 (C, S) (C,C) ≺2 (C, S)

(C,C) �1 (S,C) (C,C) (C, S)(C,C) ≺1 (S,C) (S,C) (C, S)

Table 3.3: Outcomes depend on the second-stage sustainability

From Proposition 6 we can find the SPE outcomes as indicated in the following

proposition:

Proposition 7. In the two-stage game, whether firms develop and continue to produce a

follow-on drug is determined by the sustainability of firms in the whole game and in the

second stage: SPE outcomes in the complete two-stage game can be

I. Both firms produce a follow-on drug (equivalent to (Y,Y) in the one-stage game)—

when both are sustainable in the whole game (ie. when (β − α)P > D + C and

αP > F + D + C both hold);


II. Only firm 2 produces and firm 1 stays out (equivalent to (N,Y) in the one-stage

game)—when firm 2 is the only firm sustainable in the whole game or in the second

stage (ie. either both (β − α)P < D + C and αP > F + D + C hold, or both

(β − α)P < C and αP > C hold);

III. Firm 1 engages in pre-clinical development without clinical trial, and firm 2 stays

out—when

i. firm 2 is non-sustainable in the second stage(ie. αP < C) (thus also non-

sustainable in the whole game);

or

ii. firm 2 is sustainable in the second stage but not in the whole game, and firm

1 is sustainable in the second stage (ie. all three conditions αP > C, αP <

F + D + C and (β − α)P > D + C hold).

Proof see Appendix.

The proposition makes intuitive sense: when both are sustainable in the whole game,

neither is discouraged by the prospect of sharing the follow-on market, and both stand

to gain by producing a follow-on; if firm 2 is the only firm sustainable either in the whole

game or in the second stage, it is futile for firm 1 to try to deter firm 2, plus firm 1

doesn’t stand to gain by producing, thus firm 1 stays out and firm 2 produces; in all

other cases, firm 1 can credibly threat to produce if firm 2 does, and firm 2 stands to

lose if the threat is carried out.

Notice again that I have excluded the possibility that firm 2 is unviable. As in the

one-stage game, when firm 2 is unviable, it does not enter to compete with firm 1. And

firm 1 does not have any incentive to produce a follow-on drug at all if it were not for

the purpose of entry deterrence. Hence, as long as firm 2 is unviable, the outcome would

be (N,N).


In the following Figure 3.2, as before, the directions of the arrows indicates the direc-

tions that sustainability increases.



(Y, Y ) ∼1 (N, Y )

(Y, Y ) ∼2 (Y,N)

(C,C) ∼1 (S,C)

(C,C) ∼2 (C, S)

(Y,Y)

(ES,N)

(N,Y)

a

b

b′

c

Figure 3.2: SPE outcomes depend on the sustainability in the whole game and in thesecond stage. The dashed lines (some parts coincide with the solid line) are where eachfirm is indifferent between joining a head-to-head competition and staying out in thewhole game, or where each firm is indifferent in the second stage. Each color correspondsto an equilibrium outcome. For example, the purple regions correspond to the outcome(ES,N). For a “disincentive effect” to occur, the equilibrium has to shift cross the solidkinked line.

Results I and II above are similar to those of the one-stage model, with slightly

different conditions because a firm is now allowed to abort the development process.

Comparison with Figure 3.1 is straightforward: (i) Deterrence, as long as it is success-

ful, is cheaper than in the one-stage game: in the one stage game deterrence requires fully

producing a follow-on drug (the outcome is (Y,N)) and here it only requires developing

it without production (the outcome is (ES,N)). (ii) Notice the left-middle rectangle

where deterrence occurs in the one-stage game: deterrence is no longer possible in that


region. This rectangle corresponds to the case that firm 1 is unsustainable in the second

stage, but firm 2 is sustainable in the second stage, though not in the whole game (ie.

all three conditions (β − α)P < D + C, αP > C and αP < F + D + C hold for that

rectangle). This means as long as firm 2 is the only firm sustainable in the second stage,

it can intimidate firm 1 into staying out. This is different from the one-stage model in

which as long as firm 2 is non-sustainable in the whole game it has to stay out. The

difference is due to the fact that costs are sunk at the time second stage starts in the

two-stage game. Firms have the option to quit at the start of the second stage plays to

firm 2’s advantage: it can credibly threaten to fight firm 1 in the second stage if they

both choose to enter, and firm 1, foreseeing having to quit by then, stays out.

Corollary (C5). Firm 1 cannot deter firm 2 if the latter is sustainable in the whole

game; if the latter is unsustainable in the whole game, firm 1 may deter firm 2.

Contrast this with C4, in which firm 2 being unsustainable is the necessary and

sufficient condition for deterrence.

In III, the sole purpose of firm 1’s entry is to deter firm 2. The results are different

from that of the one-stage game, since firm 1 only completes the development stage and

does not produce or market the follow-on drug. This makes more sense, given that I’ve

assumed that to firm 1, producing the lone follow-on drug doesn’t bring extra revenue

but does incur costs in each of its two stages of development. In the one-stage game, the

prediction is that as long as firm 2 is non-sustainable in the whole game, firm 1 would

produce the follow-on drug. This means firm 1 will complete the costly full development

process only to avoid the greater evil of firm 2 entering the market. Here the prediction

is more realistic: firm 1 aborts the process as soon as the objective of deterrence is

accomplished at some point during the development process.

Corollary (C6). If firm 1 chooses to deters firm 2, the former carries out first-stage

development but never the clinical trials.


So why is it always sufficient for a pioneer firm (the firm that develops the first-

in-class pioneer drug) to only deter with development and never with clinical trials?

This is because if a pioneer can credibly threaten to continue to clinical trials after

a competitor chooses to enter, and the competitor makes a loss in case both produce a

follow-on, then the competitor would not enter in the first place, because either producing

or stopping before clinical trials entails a loss. Thus it is sufficient for a pioneer to deter

with development, without having to actually continue to clinical trials. This is consistent

with the observations and drugs routinely get terminated in the process of development

for purely strategic reasons.

Another phenomenon consistent with the prediction of the model is the development

of follow-on drugs by the pioneer firm long before the expiry of the pioneer patent, when

the pioneer drug is a success.20 For example, Mevacor (see the previous footnote) by

Merck was released on the market in 1987 and was a success. It was generating sales in

excess of $1 billion when in 1991 Merck obtained approval for the follow-on drug Zocor.

While developing new drugs to replace old ones about to go off patent is a reasonable

business move and quite prevalent21, the early development seems puzzling since two

similar drugs erode into each other’s market share, and given that it’s a valid option

to release Zocor after the expiry of the patent on Mevacor, as in the practices of many

other drug companies, releasing Zocor before then seems premature. Here the model

predicts that if the “pie” is large enough, a pioneer firm will choose to enter the follow-

on market along with the competitor. It produces the follow-on early to wrestle back

20A follow-on drug can be developed by the pioneering firm or a competitor. For example, Merck’sstatin (a family of drugs that lowers blood cholesterol levels) Mevacor is the first-in-class drug, andfollow-on statins include Pfizer’s Lipitor, Bristol Myers-Squibb’s Pravachol and Merck’s Zocor. Asanother example, Captopril (trade name Capoten) is the first ACE inhibitor (a family of drugs thattreats hypertension and congestive heart failure) developed by Bristol Myers-Squibb, and its follow-onsinclude Zofenopril by the same company, Enalapril by Merck and Benazepril by Novartis.

21For example, when Prilosec (a proton pump inhibitor that treats heartburn) goes off patent inthe US, it was replaced by its follow-on drug Nexium by the same company AstraZeneca; Claritin, anantihistamine drug by Schering-Plough to treat allergies, when going off patent in 2002 was replaced byClarinex by the same company.


some market share from competitors (in Zocor’s case, Bristol Myers-Squibb’s follow-on

drug Pravachol), even if it incurs further development costs. In the case of statins like

Zocor, since their target condition, high blood cholesterol, is very common in the US

(some estimates circa 1991 put 25% for the percentage of the adult population with high

cholesterol), P is relatively large22, and the patent life is sufficiently long to induce the

equilibrium outcome (Y,Y) for statins, though it may not be sufficient for drug classes

with smaller P ’s.

3.3.3 The Disincentive Effect Revisited

In the previous one-stage model I discussed the disincentive effect of longer patents on

innovation. In this section I revisit the problem in the two-stage model, addressing the

following questions:

1. Under what conditions does an increase in patent length induce a disincentive

effect?

Such conditions unsurprisingly limit the increase in L in terms of other parameters:

As previously discussed, it is the marginal increase in L that results in shift of

equilibrium that causes this effect. But then—

2. Fix the other parameters (except P, which is proportional to L): if we look at all

marginal increases of patent length, do we always find a shift in the equilibrium

with a disincentive effect?

The answer is no. Details are in the Appendix. And this leads to the third question:

3. Suppose we increase L gradually from a very small Lmin (just large enough to keep

22Here I quote WHO burden of disease report of high income countries of the Americas from 2004.The burden of disease, as in Lichtenberg (2005), is represented by number of disability adjusted life-years(DALYs): the DALYs of Ischaemic heart disease (in which high blood cholesterol is considered a majorrisk factor) alone is greater than all infectious and parasitic diseases, all respiratory diseases (infectiousor noncommunicable) and all digestive diseases, just to give a few examples.


firm 2 viable in the whole game, ie. βLminp−F −D−C > 0) to very large. Under

what conditions can we expect to observe the disincentive effect, at least at some

point during this process?

To answer the above questions, I start with Figure 3.2 and explain that the disincen-

tive effect is caused by certain types of equilibrium shifts but not others. In short, the

equilibrium has to shift from successfully deterring firm 2 to failing to do so. In Figure

3.2, a shift of equilibrium can be represented by a shift from point a to c, or from b(b′) to

c. In the former case, there is no disincentive effect: though the SPE outcome changes at

the border line where (Y, Y ) ∼1 (N, Y ), the change in the payoff to firm 1 is continuous,

because at (Y, Y ) ∼1 (N, Y ) firm 1 is indifferent between the two outcomes. In the latter

case, there is disincentive effect: firm 2 changes from staying out at b or b′ to producing

at c, and it brings an abrupt change for the worse in the payoff for firm 1.

For the first question, the analysis in the Appendix is similar to that of the one-stage

model. Similarly, the conclusion is that the increase in patent length needs to be long

enough to change the equilibrium but not too long.

The following proposition is similar to Proposition 4 in the analysis of the one-stage

game. Notice that the equilibrium at La being (ES,N) means either

1. firm 2 is non-sustainable in the second stage (ie. αLap − C < 0)

or

2. firm 2 is sustainable in the second stage but non-sustainable in the whole game, with

firm 1 sustainable in the second stage (ie. αLap−C > 0 and αLap−F −D−C < 0

and (β − α)Lap > C):

Proposition 8. Given a patent length La and costs F , D and C that satisfy that the

equilibrium at La is (ES,N), a longer patent life Lb (Lb > La) induces disincentive effect


by switching equilibrium from (ES,N) to (Y,Y) if and only if all of the following conditions

hold:

(i) αLbp − F − D − C > 0 (firm 2 is sustainable in the whole game at Lb);

(ii) (β − α)Lbp > D + C (firm 1 is sustainable in the whole game at Lb);

(iii) Lb < (Lap + C)/[p(1 − α)] (returns to firm 1 are higher with La).

The first two conditions are identical to those of Proposition 4. The third condition

is less restrictive than in the one-stage model, because firm 1’s profit from successfully

deterring firm 2 is higher here, without carrying out clinical trials.

For the existence of Lb, the following proposition is similar to Proposition 5:

Proposition 9. Given a patent length La and costs F , D and C that satisfy that the

equilibrium at La is (ES,N), a longer patent length Lb that induces disincentive effect (by

switching equilibrium from (ES,N) to (Y,Y)) exists if and only if both [α(Lap + C)]/(1−

α) − F − D − C > 0 and (β − α)(Lap + C)/(1 − α) > D + C.

And the proof is similar to that in the one-stage model. Intuitively, La needs to be

large enough relative to sunk costs adjusted by market shares.

For the second question, the analysis in the Appendix concludes it is possible that

equilibrium shifts never change the success/failure of deterrence, thus there may not be

any disincentive effect throughout the increase of the patent length from one extreme to

the other.

For the third question, the answer is the following proposition:

Proposition 10. The disincentive effect exists if and only if either (F +D+C)/β < C/α

or (F + D + C)/α > C/(β − α).

The proof is the analysis in the Appendix. The conditions above are not quite strin-

gent: for example, if α is quite small relative to β, or if C is quite small or quite large


relative to F +D, or there is a combination of such effects, then we are bound to observe

the disincentive effect when increasing the patent life. In other words, if deterring entry

is quite profitable for firm 1, or if the clinical trial cost is highly significant or highly

insignificant, the disincentive effect occurs as firm 1 alternates from deterring firm 2 and

not being able to do so.

Intuitively, to have no disincentive effect, longer patents cannot cause a change in firm

2’s decision whether to enter: firm 2 enters the market regardless of patent length, and

for firm 1 it is futile to try to deter firm 2: either firm 2 is sustainable in the second stage

while firm 1 is not (with very short patents), or firm 2 is sustainable in the whole game

while firm 1 is not(with slightly longer patents). Firm 2 is sustainable more often than

firm 1; the condition for firm 1 to be sustainable relative to firm 2 is stricter: β −α (firm

1’s share of the market when competing head-to-head) has to be small relative to α (firm

2’s share when competing head-to-head), with costs adjusted. This is the implication of

condition (iii). Considering that firm 2 is sustainable as long as it is viable, even with

very short patents, this implies that firm 2’s share when competing head-to-head (α) has

to be large relative to that of entering alone (β), cost adjusted. This is the implication

of condition (i).

Note that the disincentive effects caused by the lengthening of the patent does not

translate to increases in the “pie”. In particular, reducing costs, despite also increasing

the “pie”, may not have disincentive effects. For example, the research cost R is unique

to the pioneer firm. Reducing R increases returns to the pioneer firm without shifting

equilibria in the game in the follow-on market. It simply encourages pioneer innovation


as expected.23

3.3.4 The Impact of the Disincentive Effect on Innovation Pro-

ductivity

Suppose given some F , D and C, when the patent length is La, the equilibrium outcome

is (ES,N), and the pioneer drug’s expected return is positive, and it gets developed.

When the patent length is Lb > La, the equilibrium is (Y, Y ) and due to the disincentive

effect, the pioneer drug’s expected return is negative, and the pioneer drug doesn’t get

developed.

Suppose also that given F , D and C ′ > C, when the patent length is La or Lb, the

equilibrium outcomes are both (ES,N). And at La, the pioneer drug’s expected return

is negative and it doesn’t get developed, while at Lb, the pioneer drug’s expected return

is positive, and it gets developed. (Since Lb > La, given the same equilibrium the latter

generates higher return for the pioneer firm.)

Compare the above two cases and consider a firm facing a choice of developing two

potential pioneer drugs, whose estimated clinical trial costs run to C and C ′ respectively.

It’s easy to see that if the patent length is La, the only drug that gets developed is the

one with the cost C. While if the patent length is Lb, the only one getting developed

is the one with the cost C ′. In the former case, the total R&D cost, R + F + 2D + C,

is lower than in the second case R + F + 2D + C ′. This provides one possible reason

why we observe the total industry R&D cost increase over the years while the number of

new drugs discovered declines. The possible reason is that increased patent length causes

23The discussion for other costs follow: (i) The development cost D is incurred to both firms beforethe second stage. Reducing D doesn’t affect sustainability in the second stage, which means that if anyfirm is unsustainable in the second stage (thus also unsustainable in the whole game), it remains so withreduced D, and there can not be any disincentive effect in such cases. (ii) The fixed cost F, unique tothe competitor firm, follows the same argument. (iii)The clinical trial cost C is incurred to both firms inthe second stage and affects the sustainability of the firms more broadly, similar to patent length, butcosts can not be reduced below certain points (certainly not below zero), which greatly limits our abilityto make any definitive prediction about their disincentive effects.


those drug classes that intrinsically require larger clinical trials be favored over others.

In other words, maybe R&D costs rise because larger projects, newly feasible due to the

increase in patent length, are favored by pioneer firms since they discourage imitators. 24

Also, notice in this case, not only do the chosen drug projects have absolutely larger

clinical trial costs, but also the relative scale of clinical trial cost as a fraction of total

R&D cost is greater. This again fits the industry facts: DiMasi et al. (1991) and DiMasi

et al. (2003)25 reveal the following: between the two studies, the total R&D cost as well as

clinical trial costs increase significantly; as well, clinical trial costs increase substantially

as a fraction of total R&D cost, from 42% to 70%26.

Indeed, with a longer patent length, the prediction of the model is as follows:

1. The average cost of new brand-name drugs goes up.

The following graph shows an example of the change in the returns to Firm 1 with

the increase in patent length. In this example, disincentive effect exists, since the

equilibrium changes from (ES,N) to (Y,Y) when patent length reaches L2. Here L2

is defined as the patent length just long enough to keep firm 2 sustainable in the

whole game, given other parameters (ie. π2Y Y (L2) = αL2p − F − D − C = 0). L1

is the counterpart to L2 for firm 1. As before, Lmin is the patent length just long

enough to keep firm 2 viable (i.e. βLminp − F − D − C = 0), in accordance with

24Similarly, drug classes with intrinsically larger development cost or fixed cost can also be chosenover those with smaller costs by the potential pioneer firm. For example, if F and C are the same,but the development cost is D′ > D. Then the equilibrium may be (ES,N) for both La and Lb (notethe firms’ sustainability in the second stage remain the same), and drug classes with D′ are chosen.However, if at La, the equilibrium is (N,Y), then a higher cost D′ may not result in an equilibrium of(ES,N) since sustainability in the second stage is unchanged by the increase in D, while an increase inC to C ′ is more likely to change the equilibrium to (ES,N).

25DiMasi et al. (1991) uses firm level data on 93 randomly selected new chemical entities introducedinto human testing between 1970 and 1982. DiMasi et al. (2003) uses firm level data on 68 randomlyselected NCEs first tested between 1983 and 1994. In both studies, costs of developing drugs thateventually fail are allocated to the successful ones.

26The costs are uncapitalized out-of-pocket in constant dollars. In the first study the costs of pre-clinical and clinical costs are 66m and 48m respectively per approved NCE in 1987 dollars; in the latterthey are 121m and 282m in year 2000 dollars. See also Scherer (2007). Neither the cause nor the impactof the four-fold increase in the clinical trial costs have been well explained in the literature.


A2 (non-triviality).

L_min L_2L_1L

Returns to Firm 1

Returns to Firm 1 with disincentive effect: example

ES,N

Y,Y

In the above diagram, the returns to Firm 1 is Lp−R−F − 2D−C, if the patent

length Lmin < L < L2, since the corresponding equilibrium is (ES,N); the return

to Firm 1 is (1 − α)Lp − R − F − 2D − 2C if L > L2, with the corresponding

equilibrium (Y,Y).

The above diagram also implies that if the equilibrium would change from (ES,N)

to (Y,Y) with the increase in patent length, then given any L and p, depending on

what cost combinations are available among potential projects, the most profitable

ones among them are either (i) projects with costs just low enough that the corre-

sponding equilibrium is (ES, N) (and if costs were slightly lower, the equilibrium

would become (N,Y) or (Y,Y)) or (ii) projects with extremely low costs, and the

equilibrium is (Y,Y). 27 The most profitable pioneer projects are of course also the

ones most likely to be chosen by the pioneer firm. If case (ii) above are the only

chosen projects, then they remain the chosen ones when patent length increases,

and the costs remain the same; yet if case (i) above are also chosen projects, the

27To see this, we keep L and p constant and look at all possible combinations of costs R, F , D and C.Obviously, if all costs are very low, L2 is very small, so the equilibrium is (Y,Y) and the returns to Firm 1is close to (1−α)Lp; if the costs are just high enough to keep L < L2 (ie. αLp−F −D−C < 0), then theequilibrium is (ES,N) and the returns to Firm 1 is Lp−R−F−2D−C = (1−α)Lp+αLp−R−F−2D−C <(1− α)Lp−R−D < (1− α)Lp. This means projects with extremely low costs and no entry deterrenceare more profitable than projects with higher costs and entry deterrence. Conversely, if extremely lowcosts are not feasible, then projects with costs large enough for deterrence will be more profitable forFirm 1 than modestly cheaper ones.


costs increase because with longer patents, the projects that are “marginal” are

more expensive. 28

Note this is independent of technological advance. The disincentive effect in said

case (i) makes large, expensive projects preferable to small, cheap ones, so even

if technological advance reduces all costs of all projects, expensive projects, newly

feasible due to the increase in patent length, will be chosen over cheap ones, driving

up the average cost.

2. The number of follow-on drugs, relative to pioneer drugs, increases.

The most profitable projects, as noted above, may correspond to equilibria that are

either (ES,N) or (Y,Y). Given a set of potential pioneer projects, if only the most

profitable ones are chosen to be pursued, then the longer the patent term, the more

likely that the most profitable projects correspond to the latter equilibrium, (Y,Y):

longer patents change the equilibrium from (ES,N) or (N,Y) to (Y,Y). This means

in the most profitable projects, increasing the patent term increases the number of

follow-on drugs per pioneer drug from zero to two. That means with an increase

in patent terms, there will be an increase in the number of follow-on drugs relative

to pioneer drugs, a phenomenon we observe in the industry.

3. The clinical trial cost rises as a proportion of total cost.

This is because choosing projects with large clinical trial cost is more effective in

deterring entry than choosing projects with other large costs, keeping the total cost

constant. Large clinical trial costs makes firms unsustainable in the second stage,

thus inducing an equilibrium in which the pioneer firm can successfully deter the

entry of the competitor. Such an effect is not achieved by increasing other costs.

All the above predictions are validated by observations in the industry, as explained

28I temporarily ignore the possibility of a shift in cases from (i) to (ii) caused by the increase in patentlength. This will be addressed in the next prediction.


in the introduction of the paper or in this section.

3.4 Discussions

3.4.1 Multi-period Modeling

With patent length implicitly modeled as an argument of market size, the above analysis

naturally raises the following question: will the disincentive effect also exist if the patent

lengths are explicitly modeled? That is, if we allow firms to respond with delay, so that

firms could potentially enter the market or start clinical trials at different times, could a

longer patent ever cause lower profit for firm 1 and discourage pioneer innovation?

As revealed in the previous two models, the disincentive effect arises if (i) firm 1

successfully preempts firm 2 when the patent life is short but (ii) fails to do so with a

slightly lengthened patent life. If the same shift of equilibrium occurs when firms can

enter at different times, then the disincentive effect continues to arise.

Here the focus is on the case in which at some short patent life Ls, firm 1 is the only

firm sustainable in the second stage if firms enter immediately (if they enter very late

apparently neither is going to be sustainable in the second stage). While with a very

long patent life, we know firm 1 isn’t able to preempt firm 2, is firm 1 able to do so with

the shorter patent life Ls?

The answer is yes.

Firm 1 can preempt if its threat of continuing to clinical trial is credible and firm

2 doesn’t want to engage in head-to-head competition. Given that firm 1 is the only

firm sustainable in the second stage if it enters immediately, the threat to continue to

clinical trial is credible if it enters immediately. In other words, as long as firm 1 enters

immediately, it is credible that it will continue to clinical trial if firm 2 also enters. This

is in fact a similar conclusion to the previous model: firm 1 only fails to preempt if firm 2

is the only firm sustainable in either the whole game or the second stage; and here since


firm 1 is the only firm sustainable in the second stage, both cases are ruled out.

While firm 1 is able to preempt, whether it chooses to do so is a different issue. This

issue is what distinguishes the multi-period model from the previous models. There is a

possibility that firm 1 doesn’t even need to bother to preempt. In particular, if firm 2’s

strategy is to develop a follow-on drug much later, e.g. shortly before the expiry of the

patent of the original drug, does this move mute the harm to firm 1, and therefore avoid

an attack from firm 1 in the form of preemption? This is an interesting question which

entails complex formal analysis. Yet we don’t need to answer this question to see that

disincentive effect exists. Whether firm 1 chooses to preempt firm 2, the fact that firm

1 is able to preempt firm 2 implies that the payoff firm 1 gets is no less than the case

where it chooses to preempt. So when the life of the patent increases marginally and

firm 1 loses the ability to preempt, its profit decreases–this is the disincentive effect.

I conclude that the disincentive effect also occurs in multi-period modeling with ex-

plicit timing. Such a model will not be analyzed in detail because the insight is sufficiently

provided in a simple two-stage model.

3.4.2 Predictions of the Two-Stage Model and Resolution of

Paradoxes

Innovation as Entry Deterrence

This model predicts that the pioneer firm may develop a drug to deter the entry of the

competitor. This accords with the observation of strategic termination of drug develop-

ment for non-medical reasons, outlined in the Introduction section.

Disincentive Effect and Innovation Productivity

Notice that the model presented here makes the simplest assumption about the relation

between R&D spending and drug discovery: as long as an adequate investment is made,


(in a pioneering drug, the total amount of investment is R + F + D + C; in a follow-on

drug, it is D + C for the pioneer firm, and an extra F for the competitor), a drug is

bound to be discovered. This should guarantee a direct link between R&D spending

and new drug discovery. Yet once we consider strategic interactions, such a link breaks

down: more R&D spending may correspond to fewer new drugs. But this is precisely

the observation from the pharmaceutical industry. Previously, this observation lacked an

analytical explanation.

Discouragement of Pioneer Innovation

The above analysis explains in detail how longer patents can reduce the returns (quasi-

rents) of the pioneer firm, which in turn discourages pioneer innovation. In short, longer

patents may attract more competition in the same drug class, reduce the profitability of

pioneer drugs and increase the profitability of follow-on drugs. This distorts the incentive

of the pharmaceutical companies, encouraging more follow-on research and less pioneer

research, and the amount of time the pioneer drug remains unique in its therapeutic class

is reduced.

Early Developed Follow-Ons by the Pioneer Firm

The model is consistent with the puzzling phenomenon of pioneering firms producing

follow-on drugs early, well before the expiry of the patent on the successful pioneer drug.

If the market size (total quasi-rent) is large, the pioneer firm can use the follow-on to

increase its current market share.

Rate of Continuation

If firm 2 enters the development stage and the drug does not fail, firm 2 will continue to

the clinical trial stage, while firm 1 may stop. This provides a testable prediction: the

data should reveal a higher ratio of continuation (number of drug continuing to clinical


trials/number of INDs) for firm 2, controlling other factors that affect the success rate

of drugs for each firm.

3.5 Conclusion

At the outset, I presented some puzzling observations in the pharmaceutical industry: de-

spite the urgent need of pharmaceutical companies to produce new drugs, promising new

drugs are cancelled mid-development for non-medical reasons; despite enhanced R&D

technology, the R&D costs have increased dramatically while the number of discoveries

have decreased; despite the increase in nominal length of exclusivity, the actual period

of exclusivity for pioneer drugs has decreased markedly.

Such phenomena are observed simultaneously. Is the concurrence simply a coinci-

dence, or are the observations related to each other? Is there a unifying explanation for

all of them? Are there any implications we can deduce from such observations? Based

on the standard entry deterrence literature, the two-stage model developed here explains

the phenomena. In this model, a firm pondering developing a first-in-class pioneer drug

realises that the existence of a pioneer drug may attract follow-on drugs developed by

competitors to the same therapeutic class. And once the pioneer drug is produced, the

pioneer firm may choose either to defend its market share or to tolerate the competition

by making two sequential decisions (whether to start, and whether to continue) in the

drug development process.

The model predicts: (i) When deterring the entry of follow-on drugs is feasible, the

pioneer firm only completes the first-stage development and never continues to the clinical

trial stage, which fits the first observation; (ii) increased patent length can cause expensive

projects to be favored by the pioneer firm, especially those involving large clinical trials,

since expensive projects are more capable of deterring entry from competitors, and this

fits the second observation; (iii) longer patent length shifts the incentive of innovation


towards follow-on research, instead of pioneer research, and decrease the amount of time

pioneer drugs remains unique in the therapeutic class, and this fits the third observation.

Aside from such observations, the model also predicts that in a large market, pioneer

firms will develop follow-on drugs to defend market share before the expiry of the pioneer

patent, which also fits the industry observation. All the above raises the following issue:

despite the fact that patent protection is essential for the survival of pharmaceutical

companies, excessively long protection may distort the incentives towards less original

research, misallocate resources and induce lower research productivity in the industry.

The above leads to other policy questions. For example, should drug prices be reg-

ulated? If so, how should these prices be regulated? Should new drugs to required to

demonstrate superiority over existing drugs? If so, how should superiority be defined?

Is it a reasonable proposition to carry out all clinical trials in a centralised institution?

These are topics of future research.


3.6 Appendices

3.6.1 Analysis of the disincentive effect in the one-stage model

I discussed that the equilibrium is (Y,N) at the patent length L = Lmin, and the equi-

librium is (Y,Y) when L is very large.

With the increase of L from Lmin, the path of the equilibrium change has two switching

possibilities: Case I is that as L increases from Lmin the equilibrium changes directly from

(Y,N) to (Y,Y); Case II is that as L increases from Lmin, the equilibrium changes from

(Y,N) first to (N,Y) and then to (Y,Y). Define Li (i = 1, 2) as the length of the patent just

sufficient to keep firm i sustainable. That is, L1 is defined by π1Y Y (L1) = (1 − α)L1p −

D−C = π1NY (L1) = (1−β)L1p and L2 is defined by π2Y Y (L2) = αL2p−F −D−C = 0.

Which switching case prevails depends on the relative scale of L1 and L2. The former

occurs if L1 < L2; the latter if L1 > L2.

In Case I, with the increase of L, firm 1 becomes sustainable before firm 2 since firm

1 requires a smaller L for sustainability. From Figure 3.1 we can see that the equilibrium

is (Y,N) before firm 2 becomes sustainable; and it is (Y,Y) after.

L_min L_1 L_2L

Returns to Firm 1

Returns to Firm 1 with the increase in patent length: Case I

Y,N

Y,Y

In Case II, firm 1 becomes sustainable after firm 2, meaning that at certain moderate

level of L, firm 2 is sustainable while firm 1 is still non-sustainable, which from Figure 3.1


we can see that the equilibrium is (N,Y). So the equilibrium changes from (Y,N) when

neither is sustainable, to (N,Y) when only firm 2 is sustainable and then to (Y,Y) when

both are sustainable.

L_min L_2 L_1L

Returns to Firm 1

Returns to Firm 1 with the increase in patent length: Case II

Y,N

N,Y

Y,Y

For the purpose of discussion I focus on Case I where there is a direct shift from (Y,N)

to (Y,Y)29.

The Case I diagram suggests that if we hold p constant, a small increase in L that

shifts equilibrium outcome from (Y,N) to (Y,Y) induces lower returns for firm 1 30, despite

the increase in the “pie” size P = pL. This is the disincentive effect of longer patent

length.

The transition in equilibrium can be represented with the following two scenarios with

patent life La and Lb (Lb > La) respectively. In the first scenario with a patent length of

La, the equilibrium outcome is (Y,N); and in the second scenario with a patent length

of Lb, the equilibrium outcome is (Y, Y ). In the first scenario, firm 2 is non-sustainable

(ie. αLap − F − D − C < 0); in the second, both firms are sustainable, (ie. both

(β − α)Lbp > D + C and αLbp − F − D − C > 0). The return to firm 1 in the first

scenario is Lap − D − C and in the second scenario the return is (1 − α)Lbp − D − C.

29In fact, as can be seen from the diagram, a shift in equilibrium from (Y,N) to (N,Y) also has a“disincentive effect”.

30If L is very large, say approaching infinity, obviously the returns for firm 1 is also going to be verylarge.


Here (Case I) generating a reduced return to firm 1 requires entry by firm 2. With La

such that αLap−F −D−C < 0 (ie. firm 2 is non-sustainable at La), entry by firm 2 with

patent life Lb requires a sufficiently longer Lb such that at Lb, αLbp − F − D − C > 0

(firm 2 is sustainable at Lb), (β − α)Lbp > D + C (firm 1 is sustainable at Lb) and

Lb < La/(1 − α) (returns to firm 1 are higher with Lb). Proposition 4 formalizes this

analysis.

For such an Lb to exist, the necessary and sufficient condition is that both αLap/(1−

α) − F − D − C > 0 and (β − α)Lap/(1 − α) > D + C hold.

We know it’s sufficient, because there must exist some Lb slightly smaller than La/(1−

α), such that if we replace La/(1 − α) in the above two inequalities, the direction of

the inequality is unaffected. This means such an Lb satisfies all three conditions in

Proposition 4, generating the disincentive effect.

We know it’s necessary, because if either of the two inequalities doesn’t hold, then

any Lb that satisfies Lb < La/(1 − α) will break either condition (i) or condition (ii) of

Proposition 4.

Proposition 5 formally states the above result.

3.6.2 Analysis of the Two-stage model

Competition in the second stage

Suppose both firms have chosen to develop a follow-on in the first stage, what decisions

about clinical trial will they make? The payoffs for continuing/stopping are summarized

in the following table. Notice that the costs of pre-clinical development have already

sunk. If a firm develops the drug, it essentially is paying for the clinical trial in exchange

for a larger share of the market.


Firm 2

Firm 1Payoffs C S

C (1 − α)P − C, αP − C P − C, 0S (1 − β)P , βP − C P , 0

Table 3.4: Payoffs in the second stage: two-stage model


Corollary (C7). By A2, if firm 1 doesn’t continue, firm 2 prefers to continue, ie.

(S,C) �2 (S, S).

Proof. βP − F − D − C > 0 ⇒ βP − C > 0 ⇔ (S,C) �2 (S, S).

This corollary says that since firm 2 is viable in producing the follow-on drug, it must

be viable in continuing (to the clinical trial). We give a formal definition of being viable

in continuing as follows.

Definition 5. Firm 2 is viable in continuing to the clinical trial if (S,C) �2 (S, S); it

is nonviable in continuing if (S,C) ≺2 (S, S).

As usual, we ignore the case in which firm 2 is neither viable nor nonviable.

Corollary (C8). By A2, firm 1 prefers to continue alone than to have firm 2 continue

alone, ie. (C, S) �1 (S,C).

Proof. βP − F − D − C > 0 ⇒ βP − C > 0 ⇔ P − C > (1 − β)P ⇔ (C, S) �1 (S,C)

As before, I rank the payoffs of firm 1 and 2 separately.

Firm 1 I. Best: neither firm continues;

II. Second best: firm 1 alone continues;

III. Third and fourth, unranked: Firm 2 alone continues/both continue(competing

head-to-head). If competing head-to-head is better for firm 1 than the situation in which

firm 2 alone continues, then we say firm 1 is sustainable in the second stage.

Whether firm 1 is sustainable in the second stage depends on

• the size of the “pie”, as captured by P ;

• 1 − α and 1 − β: how much of the market share firm 1 can keep if firm 2 alone

enters and both firms enter;

• C: the scale of the clinical trial cost.


Firm 2 I. Best: (positive payoff) Firm 2 alone continues;

II. Zero payoff: firm 2 doesn’t continue, regardless whether firm 1 does;

III. Positive, negative or zero payoff: both continue. Firm 2 is sustainable in the

second stage if the payoff of competing head-to-head is positive, non-sustainable in the

second stage if it is negative. When firm 2 is sustainable in the second stage, head-to-head

competition is better than stopping for firm 2.

Whether firm 2 is sustainable in the second stage depends on

• the size of the “pie”, P ;

• α: how much of the market share firm 2 can get if both firms enter;

• C: the scales of the costs.

If both firms chose to develop the follow-on in the first stage, then in the second stage,

whether any one would continue depends on whether they are sustainable in the second

stage.

Proof of Proposition 6

Proof. I. Since firm 2 is sustainable in the second stage, it continues to the second stage

(choosing action C) regardless of whether firm 1 continues. Firm 2’s choice of C in

turn induces firm 1 to choose C, since for firm 1, continuing (thus competing head-

to-head with firm 2) is preferable to stopping (accommodating firm 2 by allowing

it to be the sole producer of follow-on). Thus both firms continue and the outcome

is (C,C).

II. As said above, if only firm 2 is sustainable in the second stage, then firm 2 chooses

C regardless of firm 1’s choice. Since firm 1 prefers to accommodate firm 2 rather

than compete head-to-head, given firm 2 chooses C, firm 1 chooses S. Thus only

firm 2 continues, and the outcome is (S,C).


III. If firm 2 is non-sustainable in the second stage, then it only chooses C if firm 1

chooses S. But will firm 1 choose C or S? If firm 1 chooses C, it knows firm 2 will

then chooses S, and firm 1 ends up being the sole producer of follow-on; if firm 1

chooses S, it knows firm 2 will then choose C, and firm 1 ends up accommodating

firm 2. Since by corollary C1 firm 1 prefers the former, firm 1 is the only firm that

continues and the outcome is (C,S).

Proof of Proposition 7

Proof. I. when both are sustainable in the whole game, by Remark 1, we know both

are also sustainable in the second stage. By Propostion 6, the second-stage outcome

is (C,C) if both firms choose to enter in the first stage.

Since firm 2 is sustainable in the whole game, it chooses to enter in the first stage

regardless of whether firm 1 enters: if firm 1 also enters, in the second stage both

continues, so in the end both produces a follow-on; if firm 1 doesn’t enter, then firm

2 enters and continues alone–in either case firm 2 prefers entering to staying out.

Given firm 2 enters in the fist stage, firm 1 is better off entering as well, since by

the definition of sustainability of firm 1, firm 1 is better off entering and competing

head-to-head with firm 2 than staying out and accommodating firm 2.

Thus when both are sustainable in the whole game, both enter in the first stage

and both continue in the second. The outcome is equivalent to (Y,Y).

II. i. If firm 2 is the only firm sustainable in the whole game, by the analysis above,

firm 2 enters in the first stage regardless of firm 1’s choice. Also, by Remark 1,

firm 2 is sustainable in the second stage. We don’t know whether firm 1 is.

Suppose firm 1 is also sustainable in the second stage. By Propostion 6, the

second-stage outcome is (C,C) if both enter in the first stage, resulting in both


firms producing follow-ons. But since firm 1 is non-sustainable in the whole

game, it prefers to stay out given firm 2 enters regardless. Thus firm 1 stays

out and firm 2 enters and then continues. The outcome is equivalent to (N,Y).

Suppose firm 1 is non-sustainable in the second stage. By Propostion 6, the

second-stage outcome is (S,C) if both enter. Apparently, given firm 2 enters

regardless, firm 1 shouldn’t enter. Still, the outcome is equivalent to (N,Y).

ii. If only firm 2 is sustainable in the second stage, by Proposition 6, if both firms

enter in the first stage, only firm 2 continues in the second stage. So if firm

1 chooses to enter the first stage, firm 2 can choose to enter as well, and then

it will be the only firm producing a follow-on. This is better for firm 2 than

staying out. Thus firm 2 enters regardless of whether firm 1 does.

Given firm 2 always enters, firm 1 is better off staying out, because even if it

enters it wouldn’t continue, and it’s better off staying out.

Thus the outcome is that only firm 2 enters and then continues, equivalent to

(N,Y).

III. i. If firm 2 is non-sustainable in the second stage, thus also non-sustainable in the

whole game, then as long as firm 1 enters, firm 2 will stay out. This is because

if firm 2 also enters, then by Proposition 6 it will not continue, and is worse

off than staying out altogether. But if firm 1 doesn’t enter, firm 2 will enter

and continue. Firm 1 prefers the former. Thus firm 1 is the only firm that

enters, but it won’t continue as it has already preempted the entry of firm 2

and doesn’t further benefit and continuing. The outcome is (ES,N).

ii. If firm 2 is sustainable in the second stage but not in the whole game, and firm 1

is sustainable in the second stage, then as long as firm 1 enters, firm 2 will stay

out. This is because by Proposition 6, both choose to continue if both choose

to enter in the first stage, and firm 2 is worse off than staying out altogether


because it’s non-sustainable in the whole game. By the same argument as the

previous paragraph, firm 1 prefers to enter and then chooses not to continue.

The outcome is (ES,N).

3.6.3 Analysis of the disincentive effect in the two-stage model

The following are the analysis of the questions.

1. Patent length shifts that cause disincentive effect.

As we’ve seen before, for the disincentive effect, a longer patent needs to shift

equilibrium, in this case from (ES,N) to (Y,Y) 31: firm 1 fails to deter firm 2 when

the patent length increases. Intuitively, for the disincentive effect, the increase

shouldn’t be too large, which in turn implies the original length of the patent

shouldn’t be too small.

I compare scenarios with lengths La and Lb again.

When the patent length is La, and the equilibrium outcome is (ES,N); and when

the patent length is Lb, the equilibrium outcome is (Y, Y ).

In the first scenario, firm 2 is (i) non-sustainable in the second stage or (ii) sus-

tainable in the second stage but non-sustainable in the whole game, with firm 1

sustainable in the second stage. Meaning (i) αLap − C < 0 or (ii) αLap − C > 0

and αLap−F −D −C < 0 and (β − α)Lap > C. The profit for firm 1 is Lap−D.

In the second scenario, both are sustainable in the whole game, meaning (β −

α)Lbp > D+C and αLbp−F−D−C > 0. The profit for firm 1 is (1−α)Lbp−D−C.

The decline of the profit for firm 1 with the increase of patent length would mean

Lap > (1 − α)Lbp − C.

31In fact, a shift in equilibrium from (ES,N) to (N,Y) also has disincentive effect with similar intuition,but for illustration I focus on the shift from (ES,N) to (Y,Y).


Thus given that La satisfies the above conditions (i) or (ii), some Lb > La that

brings less profit to firm 1 needs to satisfy αLbp−F−D−C > 0, (β−α)Lbp > D+C

and Lb < (Lap + C)/[p(1 − α)]. A patent length shift with La and Lb satisfying

these conditions causes a shift in equilibrium from (ES,N) to (Y,Y) and also the

disincentive effect.

Such a Lb doesn’t necessarily exist either. But it exists if and only if [α(Lap +

C)]/(1 − α) − F − D − C > 0 and (β − α)(Lap + C)/(1 − α) > D + C. As before,

if La is very small, there does not exist a Lb.

2. Disincentive effects do not always exist.

The diagram 3.2 intuitively gives the scenario in which the disincentive effect

doesn’t exist despite a shift in equilibrium (represented by a shift from point a

to c). As can be seen in the next point, with some parameter values, there is never

disincentive effect because only such shifts occur with the increase of patent length.

3. The condition for the existence of disincentive effects.

Consider L increases from the small initial value Lmin that just satisfies βLp−F −

D − C > 0. At Lmin, firm 2 is certainly non-sustainable in the whole game, for

the same reason as previously argued in the one-stage model. But firm 2 may be

sustainable in the second stage (ie. αLp − C > 0) —

(A.) If at Lmin, firm 2 is sustainable in the second stage, plus firm 1 is non-

sustainable in the second stage, then the equilibrium starts at (N,Y). When L

becomes very large, the equilibrium is (Y,Y). If in the process of the increase in L,

the equilibrium doesn’t become (ES,N), then as we’ve discussed before, the profit

of firm 1 continuously increase with the increase of L, albeit at different rates in

different equilibria.

(B.) If at Lmin, (i) firm 2 is non-sustainable in the second stage, or (ii) both firms

are sustainable in the second stage, then the equilibrium starts at (ES,N) with


Lmin and ends at (Y,Y) with a large L, possibly becoming (N,Y) in the middle. As

discussed before, both the marginal changes, from (ES,N) to (Y,Y) and from (ES,N)

to (N,Y) mean a “lumpy” decline in profit for firm 1, indicating the existence of

the disincentive effect.

Thus for disincentive effect to never occur throughout, as long as firm 2 is viable in

the whole game, (i) it has to be sustainable in the second stage, (ii) firm 1 has to

be non-sustainable in the second stage, and (iii) with the increase of L, firm 2 has

to become sustainable in the whole game before firm 1 becomes sustainable in the

second stage. That means (i) any L that satisfies βLp−F −D−C > 0 must satisfy

αLp − C > 0, (ii) when βLp − F − D − C = 0, (β − α)Lp < C holds, (iii) when

(β−α)Lp > C, αLp−F −D−C > 0. This in turn means (i) (F +D+C)/β > C/α,

(ii) (F + D + C)/β < C/(β − α), (iii)(F + D + C)/α < C/(β − α) 32. Apparently,

condition (ii) is superfluous given condition (iii). These conditions are formally

stated in Proposition 10.

The diagrams with disincentive effect are similar to those presented in the one-stage

game. The following is the diagram for the case no such effects exist. Lmin, L1 and L2

are defined as before; L1(2nd) and L2(2nd) denote the minimum length of patents that

respectively keep firm 1 and 2 sustainable in the second stage.

L_2 2nd L_min L_1 2nd L_2 L_1L

Returns to Firm 1Returns to Firm 1: no disincentive effect

N,Y

Y,Y

32All the equalities are discarded because none of the marginal cases are discussed.

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