Free/Open Source vs Closed Source Software:Public Policies in the Software Market∗

Stefano Comino† Fabio M. Manenti‡

July 2004

Abstract

This paper analyzes the impact on social welfare of public policies supporting opensource software (OSS). The focus of the paper is on desktop applications such asOpenOffice and Linux for desktops. Desktop users can be divided between those whoknow about the existence of OSS, the “informed” adopters, and the “uninformed” ones;the presence of uniformed users leads to market failures that may justify governmentintervention. We study three policies: i) mandatory adoption, when government forcespublic agencies, schools and universities to adopt OSS, ii) information campaign, whenthe government informs the uninformed users about the existence and the characteris-tics of OSS, and iii) subsidization, when consumers are payed a subsidy when adoptingOSS. We show that a part from subsidization policies, which have been proved to harmsocial surplus, supporting OSS through mandatory adoption and information campaignmay have positive welfare effects. When software adoption is affected by strong net-work effects, mandatory adoption and information campaign induce an increase insocial surplus provided that they help the market to tip towards standardization.

J.E.L. codes: O38, L51, L63.Keywords: software market, open source software, mandatory adoption, informationcampaign, subsidization, network externalities.

∗Paper presented at the 2nd IIO Conference (Chicago), at the 7th Annual EUNIP Conference (Porto)and at the Industrial Organization Workshop (Naples, 2003). The seminar audiences at the Universities ofPadua and Trento are also acknowledged. We wish to thank Ines Macho-Stadler, David Waterman, DavidPerez-Castrillo, Gianni de Fraja, Benedetto Gui and Alessandro Rossi for insightful discussions. The usualdisclaimer applies.

†Dipartimento di Economia, Universita di Trento, Via Inama 5, 38100 TRENTO (Italy), Tel. (39) 0461882221, email: scomino@economia.unitn.it

‡Corresponding author: Dipartimento di Scienze Economiche “M. Fanno”, Universita di Padova, Via delSanto 33, 35123 PADOVA (Italy), Tel. (39) 049 8274238, email: fabio.manenti@unipd.it

1 Introduction

Open source software is currently one of the most debated phenomena in the software in-

dustry, both theoretically and empirically. At the most basic level, the term open source

software simply means software for which the source code is open and available. The source

code is the set of written instructions that defines a program in its original form. “Availabil-

ity” implies that anybody can obtain the software and also the source code for free (or for a

nominal fee, usually media and shipping charges or online connection charges). A software

is said “open” when its source code can be modified by everybody.1

During the last ten years, open source software (OSS hereafter) has been flourishing

particularly in the market for high-end, sophisticated users. Apache is currently the most

popular software for web servers; its market share is about 60% of the total, more than

two times larger than Microsoft, its “closed source” commercial rival.2 Other examples

of well established open source software for sophisticated users are Sendmail, the dominant

messaging service program for routing and handling email by email servers, Free/Open BSD,

a server operating system and PERL, a programming language.3

Open source software packages are also available in the desktop, mass-market segment of

the industry. Two of the most prominent examples are OpenOffice, an open source suite for

office applications and Mozilla, a browser project initiated by Netscape.4 Contrary to the

software packages developed for professional high-end users, the open source software aimed

at targeting this segment of the market have not yet gained a significant market share.5

Perhaps the most popular open source project is the operating system Linux. Linux

is somehow peculiar; it grew in importance in the server market and, more recently, as a

desktop operating system it started competing with commercial software in the market for

low-end consumers.6

1All the details related to the open source software movement can be found at the web site

www.opensource.org.2Updated figures on active web servers can be found at www.netcraft.com.3For a complete list of the most successful open source projects, see the FLOSS report (Berlecon, 2002).4The graphic software “Gimp”, often called the “free Photoshop”, also belongs to this category. Similarly,

“Chandler” is a personal information manager intended for use in everyday information and communication

tasks such as composing and reading email. A plethora of open source applications including standard office

software, is also available for Palm computers; see www.palmopensource.com.5See Berlecon (2002), Bonaccorsi and Rossi (2003) and Evans and Reddy (2002) for further evidence.6Also Linux has gained significant market shares especially in the segment of the server operating systems.

2

The open source software is attracting a lot of attention not just in the computing and

in the academic communities but, recently, also in the political arena.7 Many governments

and municipalities around the world have already taken relevant steps in favor of open

source software usually aimed at helping the software to gain a sufficient installed base to

ignite the process of adoption; other countries are actually discussing which is the best way

to intervene in the software market and how to stimulate OSS.8 Various rationales have

been put forward to justify active government policies. Open source should be supported

because it is considered being technically superior with respect to the closed one and/or

because it is more stable and secure since it allows adopters to check and fix on their own

any possible bug and to optimize systems against viruses. Moreover, since open source is

usually available at nearly zero price, another common rationale for stimulating its adoption

in public administrations is based on budget concerns.9 In a more dynamic perspective,

the open source movement would deserve to be stimulated because its open nature provides

a better “environment” to spur innovation. The availability of the source code makes it

possible for software developers to improve upon the already existing programs.

With the relevant exception of Shapiro and Varian (2003) who strongly argue in favor

of Linux adoption in public administration because of its technical superiority and of its

open nature, the vast majority of the economic literature is skeptical about justifying forms

of government intervention in the software market. According to Evans and Reddy (2002)

there is no evidence of any market failure that would justify a governmental support in

favor of open source; Schmidt and Schnitzer (2003) go even further pointing out that any

active public policy favoring open source software always harms social welfare.10 These

See the FLOSS report (Berlecon, 2002).7There is already a relatively vast literature on OSS; the main issues addressed by this literature can

be grouped as follows: i) motivations to contribute to an OSS project, see Raymond (1999), Lakhani

and von Hippel (2003), and Lerner and Tirole (2002a) among others, ii) OSS licensing, see Lerner and

Tirole (2002b) and Valimaki (2003), and iii) competition between OSS and CSS, see Mustonen (2003) and

Casadesus-Masanell and Ghemawat (2003). Kuan (2001) looks at OSS as a result of consumer integration

into production and she investigates the market conditions that favor the emergence of OSS.8A clear, non technical and comprehensive discussion of government policies towards open source software

is in Hahn (2002).9It should be noticed that these rationales for public intervention coincide with users’ preferences; ac-

cording to the FLOSS Report, (Berlecon, 2002) the tree most important features that induce the adoption

of open source software are: better access protection, higher stability and lower licence fees.10Similarly, Smith (2002), stresses that is the marketplace rather than the policy arena to create the

3

authors observe that historically governments have a poor track record in selecting the best

technology through public intervention; they suggest that the choice of winners and losers

should be left to the marketplace, while governments should intervene only to ensure a level

playing field.

We are not fully convinced by the arguments against public intervention that have been

put forward; we believe that, especially in the mass-diffusion segment of the software market,

the level playing field cannot be taken for granted. An example may help to understand our

view. Consider the market for office applications; OpenOffice, available both under Linux

as well as under Windows, has been generally recognized as an extremely powerful and

versatile application.11 OpenOffice can be downloaded for free from the internet and it is

highly compatible with Microsoft Office; however, despite all these features its use is still

restricted to a small audience. In our view, this fact can be easily explained: the existence

and the features of OpenOffice are unknown to a large fraction of the potential adopters.

This example is emblematic and we believe that it may generalize to other open source

packages: the fact that the level playing field is not necessarily guaranteed is due to the very

essence of OSS, and this also makes the open source software a peculiar commodity. Contrary

to commercial vendors who make money on developing and selling their software and who

devote a considerable share of their revenues in campaigning their products,12 open source

developers do not obtain direct monetary rewards from their efforts; as a consequence, OSS

developers have little or no resources to advertise their products and to inform the potential

adopters. The presence of a large fraction of uninformed users ignoring the availability

or the characteristics of open source packages seems an intrinsic feature of the software

market, especially of the mass-diffusion segment where individuals are mostly uneducated

and unsophisticated users.

The aim of our paper is to analyze the impact on social welfare of the various forms of

right incentives to software developers. Bessen (2002) does not have a clear point in favor or against such

policies. However, he claims that the current patent protection of commercial software may be source of

social inefficiencies.11Just to take a relevant quotation from the Washington Post: “. . . after using the Windows version of

OpenOffice for the past week and a half, I can attest that it either matches or beats Microsoft Office in

features and ease of use, at the cost of slower performance on older computers . . . ”(Rob Pegoraro, “The

office suite and lets you see past Redmond, Washington Post, May 12, 2002.”)12To give an idea about the expenses in advertising by commercial firms, in the year 2002 Microsoft spent

the relevant amount of $ 909 millions for advertising in the US market. Data taken from www.adage.com.

4

interventions that have been proposed by many governments around the world; our model

is intended to describe competition in the segment of desktop, mass-market applications

(i.e. desktop operating systems, wordprocessors, spreadsheets or other office suites in gen-

eral) characterized, as discussed above, by the presence of a large fraction of unsophisti-

cated/uninformed users.

Our analysis moves from the contribution of Schmidt and Schnitzer (2003); the set up is

based on a very simple static model in which two software products are offered. The closed

source software is supplied by a commercial firm, while the open source is sold at marginal

cost by a fringe of independent developers.

Absent quality differences between products, we assume that potential adopters can be

segmented into two distinct categories: the informed users who know about the existence

of both the closed and the open source software and the uninformed ones who ignore the

existence and/or the characteristics of the open source alternative. Clearly, competition

between open and closed source software occurs only in the market for informed users and

we model it according to a standard Hotelling-like framework.13 The formal analysis is

divided in two sections. In Section 3 we consider the basic model; Section 4 extends this

framework to take account of a relevant feature of software markets, namely the presence of

network effects. Software products are extensively affected by network externalities since the

individual benefit from using a given package increases with the number of individuals having

adopted the same software or a compatible one. By including these effects into the model we

are able to understand better the impact of government intervention in the software market.

In order to characterize the market failures, we start by analyzing the pure market equi-

librium, that is the equilibrium when the government does not intervene, and then we show

under which circumstances an active government intervention may be desirable. The re-

maining sections of the paper are devoted to present the impact on social welfare of the

most commonly proposed government policies intended to correct these market failures.

To understand current governments thinking on open source it is useful to survey the

major public initiatives to support OSS.

13Other “competitive battlefields” that may fit our framework are the following: OpenOffice vs MS Office,

Gimp vs Photoshop, Linux for desktop vs MS Windows etc.

5

1.1 Government policies towards open source

Governments are great purchasers of computer software. As reported in Evans and Reddy

(2002), the US government alone spent $3.7 billion on software in 2000. A first and direct

public intervention in the software market is therefore very simple: whatever its merits

and characteristics, mandate adoption of OSS in public administration and at schools or

universities.

Many governments have already decided to support OSS in this way: in Latin America,

the Brazilian government passed a legislation that mandates OSS to be adopted in municipal

governments; in France, the government has decided to install OSS on desktops as part of

the project aimed at computerizing much of the country’s administration by 2007.14 In other

countries, pending legislations are waiting for parliamentary approval: in Italy too a similar

legislation is currently under discussion. The City of Austin (US) in January 2004 announced

that it would migrate several hundred Microsoft Office installations to OpenOffice.15

Other governments, such as Germany and Singapore, have also adopted a different ap-

proach. Singapore is offering tax breaks to companies that use the open source Linux op-

erating system instead of Windows; in Germany, an agreement has been reached between

the government and IBM that offers discounts on IBM machines with pre-installed Linux.

In both these examples, the adoption of OSS rather than the commercial alternative is

subsidized either through tax savings or hardware discounts.

A third policy, certainly less intrusive than the first two and for this reason also very

popular around the world, is based on supporting OSS through promotional campaigns. In

the US, many public education consortiums are promoting OSS. Just to take an example,

in North Mississippi, in August 2002 the local Education Consortium has adopted a pilot

program called “freedom to learn” aimed at campaigning OSS in public schools: students

have been introduced to OSS through 2002 and 2003 school years.16

Following these examples, it is possible to distinguish the various public initiatives to-

14This project is known as ADELE. See www.IDG.net.15The Israeli government has recently announced that agencies will continue to use Microsoft Office, but

that they will stop upgrading it; at the same time the government is promoting the use of open source

alternatives such as OpenOffice.16It must be noted that all the initiatives based on mandating public schools to adopt OSS also fall in this

third category: by using OSS at schools, students learn about OSS and this certainly constitutes a form of

promotional campaign.

6

wards OSS in the following three broad categories: i) mandatory adoption, ii) subsidization

and iii) information campaign.

The rest of the paper is organized as follows: Sections 2 and 3 present the basic model

under the simplification of no network effects; Section 4 extends the analysis to the case of

network externalities. In Section 5 we generalize our results and we discuss some open issues.

2 The model

2.1 Firm’s behavior

Competition occurs between a closed source software (CSS hereafter) and an open source

software (OSS). CSS is supplied by a single commercial firm while OSS is offered by a fringe of

independent developers. Products are horizontally differentiated and we model competition

using a standard Hotelling framework; the two products are located at the extremes of a

unit length segment: CSS at 0 and OSS at 1.

Open source software is sold at marginal cost while CSS is priced at p by its producer.

We assume zero marginal production cost for the two software and that the commercial

vendor is not able to discriminate consumers according to their location or to their type.

2.2 Consumers

There are two types of consumers: a) the “informed” users, i.e. those who know about the

existence of both CSS and OSS and that take their adoption decision comparing the utility

given by each alternative, and b) the “uniformed” users, i.e. those who ignore the existence of

OSS and that, therefore, consider only the closed source software when taking their adoption

decision. We assume for simplicity that the population of consumers is of mass 1; η is the

share of the uninformed, while 1 − η is that of the informed consumers. Irrespectively on

their type, consumers are uniformly distributed on a unit length segment.

A consumer located at x ∈ [0, 1] gets a net utility from buying the closed source software

or the open source alternative of respectively:

Uc = v + m(Nc)− tx− p and Uo = v + m(No)− t(1− x). (1)

where v is the gross utility from adopting the software and m(Ni) is the network externality,

that is the individual benefit of joining the network of size Ni, with i = c, o (c = closed,

7

o = open). Finally, t is a transportation cost and may be interpreted in many ways: the

cost of learning how to use the software, the installation cost or the cost of adapting other

software applications.17

In order to simplify the exposition and to derive analytical results in the first part of

the paper we do not consider network externalities; that is, we assume that m(Ni) = 0.

Network effects are reintroduced in Section 4 where we discuss how the presence of positive

externalities alters our main findings.18

3 Pure market equilibrium, market failures and gov-

ernment policies without network externalities

Let us assume for the moment that there are no network effects: m(Ni) = 0. In order to

highlight the presence and the characteristics of market failures, and therefore the scope for

a possible government intervention, we determine the pure market equilibrium, that is, the

outcome that arises without public policies in place; we then compare this equilibrium with

the social optimum.

3.1 Pure market equilibrium

The two software packages compete only in the market of the informed users. In this case,

given the price p charged by the producer of the closed source, the indifferent consumer is the

one who gets the same net utility from the two versions of the software. Formally, equating

17The model can be easily extended to incorporate vertical product differentiation and/or switching costs.

In particular, if the commercial vendor is the “incumbent” who offers an updated version of its software and

the OSS is the “entrant”, those users who adopt the latter software may incur into larger switching costs;

this scenario can be implemented by letting the gross benefit v from adopting OSS to be reduced by the

higher cost of switching. Provided that switching costs are not too large, our model still applies to this case.

Similar arguments hold for the case of vertically differentiated products.18It is interesting to note that absent network effects, our theoretical model resembles what observed in

the pharmaceutical industry where branded drugs face competition from generic drugs. These latter, which

are usually sold at a lower price, are often unknown to patients. To the best of our knowledge there is no

theoretical study on the effect of public policies supporting generic drugs. See Mestre Ferrandiz (2002) for

a recent contribution on competition between branded and generic drugs.

8

Uc = Uo given in expressions (1), the indifferent consumer is located at xi = t−p2t

.19

Uninformed users ignore that an open source software is available and confront the utility

that they receive when adopting CSS with the zero utility of not buying any software.

Therefore the indifferent uniformed consumer is the one located at xu = v−pt

, where xu

solves Uc = 0.20

Given xi, xu and the distribution of consumers among informed and uninformed types, the

profits for the CSS producer are π = p((1− η) xi + η xu). The firm cannot price discriminate

and from the first order condition it is easy to derive the profit maximizing price:21

p∗ =(1− η)

(1− η) + 2ηp∗i +

2η

(1− η) + 2ηp∗u. (2)

As shown in the above expression, p∗ is a weighted average between the optimal prices

p∗i = t2

and p∗u = v2

that the firm would charge if it were able to discriminate consumers

according to their type and where the weights of these two prices depend on the distribution

of consumers in the two segments of the market. Clearly, the larger (resp. smaller) the mass

of uninformed consumers (η high/resp. low), the closer p∗ is to p∗u (resp. p∗i ). In other words,

when setting p∗ the closed source software producer cares more of the segment of the market

with the larger magnitude.

Equilibrium market shares are obtained by replacing expression (2) into xi and xu:22

x∗i =t + 3 η t− 2 vη

4t (1 + η), and x∗u =

2 v − t + η t

2t (1 + η). (3)

In the segment of informed consumers, CSS is adopted by all those consumers located to

the left of x∗i and OSS by the others. On the contrary, in the segment of uninformed users,

CSS is adopted by all those consumers who are located to the left of x∗u; consumers located

further away from the CSS do not adopt any software at all.

Finally, having defined the equilibrium market allocations and the price, we can derive

the surplus of uninformed and informed consumers respectively:

Su = η

∫ x∗u

0

(v − tµ− p∗) dµ, (4)

19The subscript i stays for “informed”.20The subscript u stays for “uninformed”.21The second order condition is d2π

dp2 = − 1+ηt < 0, which is clearly always satisfied.

22Note that we need to impose restrictions on the parameters in order to ensure the existence of the model.

See the Appendix.

9

Si = (1− η)

[∫ x∗i

0

(v − tµ− p∗) dµ +

∫ 1

x∗i

(v − t(1− µ)) dµ

]. (5)

Total welfare is defined as the sum of consumers surplus and firm’s profit: W = Su+Si+π.

3.2 First and second best allocations

Let us now consider the socially optimal solution, namely the allocation that a welfare

maximizing planner/regulator would obtain by assigning consumers with one of the two

software packages. It is worth distinguishing between two cases: second best and first best.

We define the second best as the allocation chosen by the regulator under the constraint

that uninformed consumers can only adopt CSS;23 this is equivalent to say that the regu-

lator decides “who is going to adopt and what” but she takes as given the distribution of

consumers between informed and uninformed individuals. In this case, given that the two

software have the same gross valuation, v, the social optimum requires the minimization of

the transportation costs in the segment of informed consumers;24 conversely, in the segment

of uninformed consumers, the social surplus is maximized when all the individuals for which

the gross valuation v is larger than the transportation cost adopt the closed source software.

Remark 1 (Second best allocation). The second best is achieved when i) the informed con-

sumers adopt CSS for x ≤ 1/2 and OSS otherwise and ii) the uniformed consumers adopt

CSS for x ≤ min{v/t, 1}.

By first best we mean the situation in which the planner can also determine the type of

each consumer. This amounts to say that the government is able to decide the distribution

of consumers between informed and uninformed.

Remark 2 (First best allocation). The first best is achieved when i) all consumers are in-

formed (η = 0) and ii) those located at x ≤ 1/2 adopt CSS, while they adopt OSS otherwise.

23This definition has been provided by Mas-Colell, Whinston and Green (1995), page 819.24It can be easily verified that in order to ensure the existence of the model we need to impose that v > t

2 ;

in this case efficiency requires the segment of informed users to be fully covered.

10

By choosing η = 0, the planner is able to replicate any possible allocation that it can

be obtained when η > 0. Therefore, when there are no uninformed consumers, the social

welfare is at least as large as when η > 0. Moreover, it is socially optimal to allocate all the

(informed) consumers in a way that minimizes the transportation costs. As for the second

best, this occurs when consumers are equally split between the two products: those to the

left of 1/2 adopt CSS, while the remaining adopt OSS.

3.2.1 Market failures

Market failures can be easily determined by simple comparison of the social optimum with

the pure market equilibrium given in (3); our set-up encompasses three different failures,

which are summarized in the following remarks.

Remark 3 (Market failure 1). Too few informed consumers adopt CSS: x∗i < 1/2.

As already discussed, in the segment of informed consumers, social optimality would

require to split consumers equally between the two products in order to minimize trans-

portation costs. However, market forces do not lead to this outcome: since CSS is supplied

at p∗ > 0, while OSS is available for free, the latter is adopted by “too many” informed

consumers.

Remark 4 (Market failure 2). Conditional on the presence of uninformed consumers, η > 0,

too few adopt CSS, x∗u < min{v/t, 1}.

This failure relates to the monopolistic segment of the industry, that is that of uninformed

consumers. As for standard monopoly setting, too little amount of CSS is sold: there is a

deadweight loss due to the presence of a monopolistic producer.

Remark 5 (Market failure 3). Uninformed consumers do not make socially optimal adoption

decisions.

11

This failure is driven by the fact that uninformed consumers are unaware of the existence

of OSS. Among these consumers, those who would have optimally adopted OSS (i.e. the

individuals located close to the location of the open source software) either do not buy any

software at all or do adopt the wrong one. This means that due to their ignorance about the

existence of OSS, some uninformed consumers do not make the socially optimal adoption

decision and this leads to a market failure.

3.3 Government policies towards OSS

As we have discussed in the introduction, government policies towards OSS may take essen-

tially three forms. On the one side, governments may force agencies, schools or universities

to adopt OSS. Alternatively, a less intrusive policy may be based on informing people about

the existence and the characteristics of OSS. Finally, the government can subsidize the adop-

tion of the software open source through monetary incentives. We define the first policy as

mandatory adoption, the second as information campaign and the third as subsidization.25

The first two policies have the effect of changing the mass of consumers belonging to

the two types of informed and uninformed, while the third one has the effect of increasing

consumers valuation for OSS. Formally, we model these policies in the following way:

1. mandatory adoption: the government randomly selects an amount β ∈ (0, 1) of con-

sumers and forces them to adopt OSS.26 It is realistic to assume that, as for the closed

source producer, also the regulator ignores the location (i.e. the preferences) of each

consumer and also that she cannot distinguish between informed and uninformed users.

As a consequence, when β consumers are mandated to adopt OSS, the mass of informed

users reduces to (1−β) (1−η), while the mass of uninformed users reduces to (1−β) η.

2. information campaign: the government informs a share α of the uninformed consumers

25Clearly, the government may opt for a “policy mix” by combining two or more of these policies. Fur-

thermore, also without an explicit policy mix, both subsidization and mandatory adoption have intrinsically

an informative nature. Never the less, the aim of the paper is to study each policy in isolation from the

others, in order to disentangle their “plain” welfare effect. The evaluation of a policy mix is therefore a

simple combination of the results that we obtain for each policy and it can be easily derived.26More precisely, we consider the case in which the government imposes to these consumers to adopt OSS

provided that they get a positive net benefit (v − t(1− x) ≥ 0); in case v − t(1− x) < 0, they do not adopt

any software at all.

12

about the existence and the characteristics of OSS. As a consequence, the mass of

uninformed consumers becomes η − α, while that of the informed ones increases to

1− η + α.

3. subsidization: the government pays a subsidy s to every individual who adopts OSS.

Therefore, the gross consumer’s valuation of OSS becomes v + s.27

3.4 Mandatory adoption

Mandatory adoption is the most popular policy adopted by governments around the world.

With this policy the government forces some or all the public agencies, universities and

schools to adopt OSS.

The mass of informed consumers reduces to (1−η) (1−β), that of uninformed consumers

reduces to η (1−β), while β represents the mass of mandated consumers. Clearly, this policy

has the effect of reducing the mass of potential CSS adopters. However, the proportion of

informed to uninformed consumers remains unchanged. This fact implies that the weights

in expression (2) do not change and that the policy does not alter the equilibrium price p∗

nor the equilibrium indifferent consumers defined in expressions (3).

The following proposition summarizes the impact of mandatory adoption on total welfare.

Proposition 1. If the mass of uniformed consumers is sufficiently large, then mandatory

adoption increases social welfare. Formally there exists a unique η ∈ (0, 1) such that:

when η > η,dW

dβ> 0.

According to Proposition 1, the efficacy of the policy is driven by the presence of a large

fraction of uniformed users. When η is large, there are many individuals that do not buy any

software; let us call them as “inactive” users (i.e. those that do not know the existence of

the open source software and that at the same time do not buy the commercial software); by

mandating some these users to buy OSS, which is sold at a price of zero, the policy “forces”

them to enjoy a positive benefit. Clearly, the larger η, the more “inactive” individuals are

mandated to adopt OSS and the larger the total social benefit.

27A part from subsidization, we assume that the policies are implemented at zero cost. The introduction

of an implementation cost is straightforward.

13

Note that since the equilibrium price is unchanged, those individuals that are not man-

dated to adopt OSS receive the same amount of surplus while the mandated informed con-

sumers are clearly worse-off: the policy imposes them to adopt OSS and this restricts the

set of possible packages on which they can choose. Finally, since the mass of CSS potential

adopters is reduced, also the firm is harmed by the policy.

Summing up, mandatory adoption enhances social welfare when the positive effect on

the “inactive” individuals dominates the negative effect on the informed consumers and on

firm’s surplus. This occurs when the mass of uninformed consumers is sufficiently large.28

Proposition 1 is of particular relevance. Mandatory adoption is the most common policy

towards OSS and our set-up fits what observed in reality: as discussed in the introduction,

the presence of a large fraction of uninformed users who ignore the existence of OSS is

typical of the desktop mass-market segment of the software industry. Therefore, Proposition

1 suggests that promoting OSS applications through mandatory adoption is indeed a welfare

enhancing policy.

3.5 Information campaign

Through an information campaign, some uninformed consumers receive information about

the existence and the characteristics of OSS. Formally, this policy consists in moving a

fraction α ∈ [0, η] of consumers from the uninformed to the informed segment of the market.

As a consequence, the mass of consumers that remains uninformed reduces to η−α, while that

of informed becomes 1− η + α. In our analysis we assume that the government implements

such policy at no cost.

The study of this case closely resembles the one conducted when presenting the pure

market equilibrium. However, as the policy alters the proportion of informed to uninformed

consumers, the optimal price charged by the commercial producer changes. In particular,

since the proportion of informed consumers becomes larger, (2) converges towards p∗i .

The next proposition characterizes the effect on total welfare of the information campaign:

Proposition 2. Supporting the open source software through a costless information campaign

always increases welfare.

28This result contrasts with Schmidt and Schnitzer (2003) where mandatory adoption always hurts society;

the reason for this difference is that these authors do not consider the presence of uninformed consumers.

14

As for the case of mandatory adoption, the effect of the policy on producer’s surplus is

straightforward: since the policy shrinks the captive market of uninformed users, it undoubt-

edly harms the firm. To understand the effect of the information campaign on consumer’s

surplus we need to distinguish between two cases depending on whether the price charged for

the closed source software decreases or increases with α.29 Suppose that p∗ decreases with

α; in this case consumers are better-off. Consider the various types of users; while informed

consumers who adopt CSS benefit from a lower price, the policy does not have any impact

on those who adopt OSS. Similarly, the uninformed consumers who adopt CSS benefit from

a reduction in p∗ while those who do not adopt any software are unaffected. Finally, all the

uninformed consumers who become informed as a consequence of the campaign, are better-

off. If they adopt CSS they are charged a lower price, while if they adopt OSS they benefit

from a choice that was not available before.

From this discussion it is clear that when the policy induces the firm to charge a lower

price, the information campaign makes the market working better in all respects; each market

failure is lessened by the policy: it increases the share of consumers having adopted CSS

both in the segment of informed and in the segment of uniformed consumers (failures 1 and

2) and, obviously, it induces some previously uninformed users to take a more appropriate

decision (failure 3).

Furthermore, the information campaign always increases welfare, also when it induces

an increase in the price of the closed source software. In this case, market failures 1 and 2

become more severe but these negative effects are still dominated by the positive effect on

market failure 3. In other words, the fact that an additional fraction α of consumers is now

able to make a more informed adoption decision makes the proposed policy always welfare

enhancing.

3.6 Subsidization

The last policy under scrutiny is subsidization of OSS adoption. We model this policy by

assuming that the government pays a monetary transfer s to each individual who adopt the

open source software; formally, this amounts to assuming that the gross utility from adopting

OSS becomes v + s. Moreover, we also assume that the cost of the subsidy is entirely borne

29Since, as α increases, the optimal price converges to p∗i , then p∗ increases when p∗i > p∗u and decreases

otherwise.

15

by the society through lump sum taxation.30 The following proposition can be easily proved:

Proposition 3. Subsidizing the open source software always reduces welfare.

The policy induces the CSS producer to charge a different price for its software. Nev-

ertheless, to get more intuition of Proposition 3, consider first what happens if p∗ does not

change. The utility of all those consumers who adopt OSS both with or without the subsidy

increases of an amount s. However, this positive effect is exactly offset by the increase in

government expenditures and the net welfare effect is therefore zero. Similarly, also for those

consumers who adopt CSS there is no effect on social welfare: their utility does not change

nor the government pays any subsidy to them. On the contrary, for those consumers who are

induced by the subsidy to adopt OSS rather than CSS, the net effect on welfare is strictly

negative. The increase in the utility of these consumers does not compensate the increase in

government expenditure: without subsidies, CSS is their preferred software and when they

choose OSS instead, they certainly increase their utility but of an amount smaller than s.

On the top of these arguments we need to consider the effect of the subsidy on the optimal

price charged by the producer. It is possible to show that the firm, in order to protect its

market share, reacts to the subsidy by cutting the price. In terms of market failures, while

subsidizing OSS does not impact type 3 market failure, it mitigates failure 2 and it worsens

failure 1. Given that the subsidy induces the commercial software vendor to lower its price

then the deadweight loss in the uninformed segment is reduced. However, it is possible

to verify that the reduction in p∗ does not entirely compensate the subsidy payed to OSS

adopters and, as a consequence, OSS market share enlarges. This clearly accentuates market

failure 1.31

4 Public policies with network externalities

The above analysis has been conducted absent network effects. The presence of network

externalities is a recognized feature of software products and it is therefore interesting to

extend the model to take these effects into account.

30For simplicity, we assume that taxation does not have additional distortionary effects.31It is worth noticing that as long as the subsidization policy disseminates some information on the

existence of OSS, then the negative welfare effect shown in Proposition 3 is mitigated.

16

With network externalities the individual benefit from adopting a software is positively

affected by the number of other individuals having adopted the same software or a compatible

one. This effect lies on the simple observation that the more widespread a package, the easier

is to exchange files and information with other users and, therefore, the higher the utility

from adopting that particular software.32

We will consider the case of incompatible software so that the network effects associated

to one package only depend on the number of consumers adopting that particular package.33

The analysis will be conducted under the simplifying assumption of linear externalities;

formally, in expressions (1), this amounts to assume m(Ni) = θNi, where Ni corresponds to

the number of users of software i and θ ≥ 0 represents the strength of the externality.34

We proceed in the same way as in the previous section and we start by discussing the

socially optimal allocation; this will constitute the reference point for the subsequent analysis

of the welfare impact of the various policies towards OSS.

Proposition 4 (Second best with network effects). Compared to the case of no network

effects, with linear externalities the second best is achieved with more users assigned to CSS

both in the informed and in the uninformed segment of the market.

This proposition has a clear explanation. In the second best scenario, the planner takes

the mass of uninformed consumers as given. According to expressions (1), the presence of

network effects makes each software more valuable to consumers; this implies that in the

uninformed segment of the market it is optimal to have more users being assigned to CSS

32Despite the relevance of network externalities has been commonly accepted in software markets, few

econometric papers have tried to prove empirically the role of network effects in the software industry. See

Gandal (1994) and Brynjolfsson and Kemerer (1996).33We omit the analysis of compatibility for two different reasons: i) since under compatibility network

effects are the same for the two packages, the analysis closely resembles the one provided in the above section

and therefore it is of less theoretical interest and ii) the model with compatibility necessarily builds on ad-

hoc assumptions about the characteristics of network effects for uninformed users. We make the analysis of

compatibility available upon request.34Formally, Nc = (1 − η)xi + ηxu and No = (1 − η)(1 − xi). The theoretical literature on network

competition is extremely developed; often, the individual decision to adopt a given technology is assumed

being affected by his/her expectations on future installed base (see Katz and Shapiro, 1985). We follow a

simpler approach without consumers’ expectations, as in Shy (2001). Note also that the functional form for

the networks effects that we use together with the fact that we are not considering consumers’ expectations

guarantee that the equilibrium is unique.

17

than in Remark 1.

The larger second best market share for the commercial application in the segment of

informed users has a distinct explanation and it relies on the difference in the potential

installed base of the two packages; with uninformed users that can only be assigned to CSS,

the closed source software has a larger potential installed base. Therefore, to take advantage

of the larger network externalities associated to the commercial package, the government

finds it optimal to assign this software to a share of informed consumers which is larger than

the share that characterizes the second best without network effects. In other words, in the

segment of informed users, the regulator is no longer comparing two packages characterized

by the same gross utility as in Section 3, but it needs to take into account that CSS guarantees

a higher utility due to its larger installed base.35

Summing up, Proposition 4 shows that when the industry is affected by positive network

effects, it is optimal to have the market tipping towards standardization on one software.

Unfortunately, even in this simplified framework, we are not able to formally characterize

the impact of the various policies on social welfare.36 For this reason we need to resort to

numerical simulations in order to solve the model; the following sections are devoted to

present the numerical results for the three policies considered.

4.1 Mandatory adoption

In Figures 1 and 2 we plot the social welfare as a function of β for different degrees of the

network externalities. As it will be clear in a while, the amount of uninformed users, η, plays

a crucial role in the analysis; for this reason we consider two distinct market segmentations:

one with few and one with many uninformed users.37

35It can be verified, and the intuition goes straight, that the second best market shares of CSS are increasing

with the strength of the externality. Furthermore, when network effects are very strong, the second best

requires standardization towards CSS. Note also that in the first best scenario η = 0 and therefore the

potential installed base of the two software is the same: 1; this implies that when θ is sufficiently large, first

best is achieved with full standardization towards either CSS or OSS. A formal proof of these statements is

available upon request from the authors.36For the sake of completeness in the appendix we provide the formal procedure applied to derive the

welfare function with network externalities. The appendix focuses on the case of mandatory adoption; the

derivation of the social welfare for the other cases is similar and it is omitted for brevity.37All the plots shown in these sections have been drawn for v = 1.8 and t = 1.45; for each policy we have

considered the case of few uninformed consumers, η = 0.1, and the case of many uninformed consumers,

18

As shown, W (β) shifts upwards as θ gets larger: for any β, social welfare is higher the

stronger the externality due to the larger benefit that each individual obtains from software

adoption. Not surprisingly, with small network effects (θ low), Proposition 1 still applies:

the policy of mandating OSS adoption is welfare increasing when the share of uninformed

consumers is large and it is decreasing otherwise; as shown in Figures 1 and 2, for low levels

of θ, when η is small the welfare function is negatively sloped, while when η is large, W (β)

points upwards.

Fig 1: W (β) for low η Fig 2: W (β) for high η

The case with large network effects is more interesting. In line with the analysis of

the second best provided above, the general message is that the government intervention

increases welfare when it helps the market to tip towards the adoption of only one software.

Consider first the case of small η. In this scenario, the vast majority of consumers

is informed and, given that it is provided for free, OSS gains the largest users’ installed

base even without any policy. Under these circumstances, the effect of mandatory adoption

is therefore to enlarge even further the installed base of OSS and this increases the overall

η = 0.8. Finally, six different values of the strength of network effects have been considered, namely θ = 0.01,

0.05, 0.1, 0.2, 0.3 and 0.4. All these are reasonable values of the parameters; we have run a complete scan

of the parameters space and we have obtained plots with the same characteristics.

19

network effects and the welfare consequently. This is shown in Figure 1 where W (β) becomes

positively sloped for large values of θ.

When the number of uninformed consumers is large enough (η high), then, absent gov-

ernment intervention, CSS is the software with the largest installed base. In this case, the

impact of the policy depends on the amount of consumers that the government mandates to

adopt open source. When β is small, mandatory adoption reduces the market share of the

package with the largest installed base and this goes clearly against social welfare. On the

contrary, if β is sufficiently large, the policy becomes welfare enhancing: when the govern-

ment mandates many consumers to adopt OSS, then it makes open source as the software

with the largest installed base; this favors standardization and improves welfare. These

arguments explain why W (β) becomes U-shaped in Figure 2.

4.2 Information campaign

The analysis for the information campaign follows the same arguments. As shown in Figures

3 and 4, when network effects are small Proposition 2 is still valid: a costless information

campaign is welfare increasing (graphically, W (α) is positively sloped).

Things change with large network effects. Through an information campaign the govern-

ment makes some uninformed consumers aware of the existence of OSS and this increases the

market share of the open source software. As before, the policy enhances welfare provided

that it induces the market to tip towards OSS.

This always occurs when η is small since OSS is already the software with the largest

installed base (see Figure 3); when η is large, it is CSS rather than OSS to have a larger

installed base and W (α) becomes U-shaped. If the campaign reaches few uninformed con-

sumers it harms social welfare since it goes against standardization; on the contrary, if α is

large enough, then the campaign makes OSS the software with the largest installed base and

it makes the society better off.

20

Fig 3: W (α) for low η Fig 4: W (α) for high η

4.3 Subsidization

The discussion conducted for the cases of mandatory adoption and information campaign

suggests that a policy supporting OSS has a positive impact on social welfare when network

effects are strong and the mass of uninformed consumers is small. This is no longer true

when OSS adoption is supported through monetary rewards: Proposition 3 still applies no

matter the extent to which consumers’ utility is affected by network effects.

As clear by now, when η is small then open source has the larger installed base and the

promotion of its adoption through subsidization enlarges network externalities. However,

as already stressed in Section 3, for all the informed users that are induced to switch from

CSS to OSS because of the subsidy, the increase in the utility is strictly smaller than the

government expenditure. Clearly, this negative welfare effect is higher precisely when η is

small, that is when the number of (informed) consumers who can take advantage of the

subsidy is large; for this reason even when the mass of informed users is large, a subsidy

towards OSS adoption does not increase welfare.

21

Fig 5: W (s) for low η Fig 6: W (s) for high η

5 Open issues and final remarks

In this paper we have analyzed the impact on social welfare of the three most popular

government policies in support of open source software that have been proposed in many

countries. The analysis has been conducted using a static model in which an open source

and a closed source/commercial software compete.

Our analysis is intended to describe competition in the mass-diffusion segment of the soft-

ware industry where users are often unsophisticated and, therefore, generally little informed

about the existence of open source alternatives to the most popular commercial software.

Moving from this observation, we have divided consumers according to their knowledge of

the open source: some consumers know the existence and the characteristics of the software

open source while others ignore it; in the first case they are said “informed”, while they are

“uninformed” otherwise. Competition occurs only in the segment of informed consumers.

The presence of uninformed consumers leads to market inefficiencies that may justify

government policies. We have started the analysis by considering a stylized software market

without network externalities; in this scenario, we have shown that a part from promoting

OSS through public subsidization, stimulating the adoption of open source desktop appli-

cations either through information campaigns aimed at informing the uninformed users or

by mandating OSS adoption in public agencies, schools and universities may have positive

22

welfare effects.

The model is then extended to take account of positive network effects. Through numer-

ical simulations, we have shown that with small network effects the conclusions of the basic

model are confirmed. When network effects get stronger, things may change; we have shown

that supporting OSS either through mandatory adoption or information campaign may in-

crease welfare provided that the public intervention helps the market to tip towards OSS.

Applying this analysis to the mass diffusion segment of the software market, characterized

by a large fraction of uninformed users, we have shown that with large network effects a

policy which stimulates OSS needs to be sufficiently drastic to induce welfare improvements.

The aim of our analysis has been to provide a simple and useful benchmark to evaluate

the social impact of the various policies in favor of OSS. The simplicity of our model rep-

resents its main weakness since it comes at the cost of neglecting some important features

of the real world. The most important limitation of our model is certainly its static na-

ture; we do not consider neither R&D activity from software producers nor firms entry and

compatibility strategies, which have all dynamic nature.38 Nevertheless, even in the absence

of dynamics, we believe that our model represents a useful starting point for analyzing the

static welfare effects of the various policies in the mass-diffusion segment of the market. As

we have pointed out in the introduction, many of those authors who argue against active

policies in the software market base their claims on the observation that governments are

often prone to make mistakes in “picking up the winner”. This is an important aspect

that we have implicitly ruled out by assuming that CSS and OSS are qualitatively identical

and, more importantly, by assuming that the government perfectly observes the quality of

the various software. This is clearly a relevant assumption but we believe that it has a

practical justification: the mass-diffusion segment of the market is typically made of unso-

phisticated/uneducated users who devote little time in searching for the available packages

and/or in collecting information on their characteristics. The assumption that government

agencies have some deeper information seems therefore plausible.

In relation to the widespread view that governments should intervene only to guarantee

a level playing field rather than selecting the (wrong) winner, it is interesting to note that in

some sense, our results go also in this direction: of the three policies presented in this paper,

38Whether public intervention in favor of open source encourages innovation it is still an extremely con-

troversial issue. See Shapiro and Varian (2003) and Schmidt and Schnitzer (2003) for a discussion.

23

the information campaign is the less intrusive one and, at the same time, it is the policy that

performs better in terms of welfare. By informing the individuals about the existence of OSS,

the government does not select the winning technology but she simply makes consumers in

the best position to choose, leaving them free to adopt their favorite software.

Finally, it remains to be noticed that the presence of a fraction of uninformed users is not

the only possible explanation for the poor performance of open source packages in the mass-

diffusion segment of the software market. As discussed,39 due to the presence of switching

costs, consumers may be locked-in to the dominant commercial software or, more simply,

there may be some degree of vertical differentiation between alternative products. Similarly,

a potential user might be prevented from adopting an open source software because it does

not come with adequate support services. However, we are convinced that the lack of relevant

information on open source products is an important part of the picture in the mass-diffusion

segment, even though it is not the unique explanation for its poor performance.

39See footnote 17.

24

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26

Appendix

Pure market equilibrium: restrictions on the parameters. In order to ensure the existence of the

model, we need to check when the parameters satisfy the following conditions:

0 ≤ x∗i ≤ 1, 0 ≤ x∗u ≤ 1, U∗c = v − tx∗i − p∗ ≥ 0.

where p∗, x∗i and x∗u are given in expressions (2) and (3).

The first two conditions guarantee that at the market equilibrium the indifferent consumers lie

on the [0, 1] segment; the third condition ensures that the indifferent informed consumer gets non

negative utility. From the first condition derives that: v ≤ t(1+3η)2η , while from the second one:

t(1−η)2 ≤ v ≤ t(3+η)

2 . Finally, the indifferent informed consumer enjoys a non negative utility if:

v ≥ t(3+η)2(2+η) . Putting all these conditions together it is easy to verify that the model exists for:

v ∈ (v, v), where v =t(3 + η)2(2 + η)

, and v =t(3 + η)

2. (6)

Proof of Proposition 1. When the government extracts β consumers from the entire population and

forces them to adopt OSS, firm’s equilibrium profits are simply (1 − β) times the profits earned

without public intervention: π = (1−β) p((1− η) t−p

2t + η v−pt

); the optimal price is the same as in

the pure market equilibrium. Moreover, also the range of existence of the model is as in the pure

market equilibrium case.

Two cases must be considered: i) the gross utility v is not big enough and, among the mandated

consumers, only those located “non too far” from the right end of the segment adopt OSS, and

ii) v is so high that, irrespective of their location, all the selected consumers buy OSS. Formally:

i) v < t such that only those located in ( t−vt , 1) adopt OSS, and ii) v ≥ t. Note that v ∈ (v, v) so

that case i) is defined only for v ∈ (v, t) while case ii) is defined for v ∈ (t, v).

The net consumer surplus of the informed, the uninformed and the mandated consumers are re-

spectively:

Si = (1− η)(1− β)

[∫ x∗i

0(v − tµ− p∗) dµ +

∫ 1

x∗i(v − t(1− µ)) dµ

],

Su = η(1− β)∫ x∗u

0(v − tµ− p∗) dµ,

Sm = β

∫ 1

max{0, t−vt}(v − t(1− µ)) dµ.

27

where in the expression for Sm we have incorporated the two possible cases i) and ii). Let us

prove the proposition for case i); the proof for case ii) goes in the same way and it is omitted

for brevity.40 In this case, total welfare which is defined as the sum of consumers’ and producer’s

surplus, W = Si + Su + Sm + π, is:

W =((4(1− β)η2 + 8(η + β))v2 + t(1− β)(1− η)(4(3η + 4)v − t(3η + 5)))

16(1 + η)t.

Differentiating this expression with respect to β:

dW

dβ=−(3t2 − 12vt + 4v2)η2 − 2t(t− 2v)η − 16vt + 8v2 + 5t2

16 (1 + η) t.

The sign of this expression is given by the sign of the numerator which is a parabola in η, with

η ∈ (0, 1). It is easy to verify that at the extremes η = 0 and η = 1:

sign

{dW

dβ

} ∣∣∣∣η=0

= sign{8v2 − 16vt + 5t2}, sign

{dW

dβ

} ∣∣∣∣η=1

= sign{4v2}.

From simple inspection, it is immediate to check that at η = 0, dW/dβ < 0 for any v ∈ (v, t)

while at η = 1, dW/dβ > 0. This is sufficient to prove that there exists a unique value

η ∈ (0, 1) such that dW/dβ < 0 for η < η and dW/dβ > 0 otherwise.

Proof of Proposition 2. Firm’s profits are: π = p((1− η + α) t−p

2t + (η − α)v−pt

). Solving the first

order condition, firm’s optimal price is:41 p∗ = t (1−η+α)+2v (η−α)2 (1+η−α) .

Unlike mandatory adoption, with an information campaign firm’s optimal price changes with re-spect to the pure market case: we need to find the new range of parameters for the existence of themodel. It is possible to show that in this case, the model exists for v ∈ [v, v], where v = t(3+η−α)

2(2+η−α) ,

v = t(3+η−α)2 .

Replacing p∗ into the expressions for the indifferent consumers xi and xu and into the consumerssurpluses given in (4) and (5), we finally derive the total welfare:

W =4 (α− η) (α− 2− η) v2 − 4t

(η − 4− α− 6ηα + 3η2 + 3α2

)v + t2 (1− η + α) (3α− 3η − 5)

16t (1 + η − α). (7)

Differentiate with respect to α and get:

dW

dα=

(3 t2 + 4 v2 − 12 vt) (α− η) (α− 2− η) + 8 v2 + 7 t2 − 20 vt

16t (1 + η − α)2 . (8)

40Proof available on request.41The second order condition is d2π

dp2 = − 1+η−αt < 0 and it is always satisfied.

28

This function decreases with α; to see this, take the second order derivative d2Wdα2 = − (v−t)2

2t(1+η−α)3,

which, for α ∈ (0, η), is always negative: dW/dα is monotonically decreasing in α. In order

to verify that the welfare increases with α it is enough to show that dW/dα > 0 at α = η.

From expression (8): dWdα

∣∣α=η

= 20 vt−8 v2−7 t2

16t.

The sign of this expression is given by the sign of the numerator which is concave in v. It

is easy to verify that the numerator is positive at v = v and at v = v, which is sufficient to

guarantee that it is positive for any admissible value of v.

Proof of Proposition 3. Consider the market for the informed consumers. When the government

pays a subsidy s to all those individuals who adopt OSS, the indifferent consumer is located at xi,

where xi solves the following condition: v − p− txi = v + s− t(1− xi). The indifferent consumer

is therefore located at x∗i = t−p−s2t .

The location of the indifferent uniformed consumer does not change with respect the pure market

equilibrium and it is defined in (3). Firm’s profits are therefore: π = p((1− η) t−p−s

2t + η v−pt

).

Solving the first order condition, the optimal price is:42 p∗ = (t−s)(1−η)+2 η v2+2 η .

Replacing this price into the expressions for the indifferent consumers xi and xu and into the

consumers surpluses given in (4) and (5), we can derive the welfare as a function of the policy

parameter s. Moreover it is possible to verify that: dWds = − (1−η)(s+t(1−η)+7 sη+2 η v)

8t(1+η) < 0.

Proof of Proposition 4. The Proposition can be easily proved. In the second best scenario the

planner takes η as given and chooses the allocation (xi, xu) in order to maximize total welfare. Let

us define Bc(θ, x) = v + θNc − tx and Bo(θ, x) = v + θNo − t(1 − x) as the individual benefit of

adopting CSS and OSS respectively.

The second best market share in the segment of uninformed users is xsbu (θ) = min{1, x} where

x solves Bc(θ, x) = 0. Since, by construction, Bc(θ, x) > Bc(0, x), ∀ θ > 0 and ∀x ∈ [0, 1],

then xsbu (θ) ≥ xsb

u (0) ∀ θ > 0: with positive network effects, the second best is achieved with more

uninformed users assigned to CSS. Similar arguments apply to the informed segment of the market;

in this case, two scenarios are possible: either xsbi (θ) ∈ {0, 1}, namely there is a corner solution, or

xsbi (θ) ∈ (0, 1), namely the second best is an interior. In the first case, it cannot be that xsb

i (θ) = 0:

provided that there are uninformed users and these are assigned to CSS, then, due to larger installed

42The second order condition is d2πdp2 = − 1+η

t < 0 and it is always satisfied. Moreover, the model exists for

v ∈ [v, v], where in this case: v = (t−s)(3+η)2(2+η) , and v = t(3+η)−s(1−η)

2 .

29

base, assigning all informed consumers to CSS implies a strictly larger welfare. In the second case,

optimality requires indifference between CSS and OSS: Bc(θ, xsbi (θ)) = Bo(θ, xsb

i (θ)), ∀ θ ≥ 0.

Again, provided that there are uninformed users assigned to CSS, Bc(θ, 1/2) > Bo(θ, 1/2); this

implies that xsbi (θ) > 1/2 ∀ θ > 0 and it completes the Proof.

The Welfare function with network externalities. 43 In this appendix we show the procedure ap-

plied to derive the welfare function with mandatory adoption when the individual utility is affected

by network effects as in (1). We focus on the case with v > t so that all the consumers that are

mandated by the government do actually adopt OSS. A similar reasoning applies for v < t.With network effects the location of the indifferent informed and uninformed consumers are obtainedby solving the following system of equations:

{v + θ(xi(1− η)(1− β) + xuη(1− β))− txi − p = v + θ((1− xi)(1− η)(1− β) + β)− t(1− xi),

v + θ(xi(1− η)(1− β) + xuη(1− β))− txu − p = 0

where the first equation defines the indifferent informed consumer while the second one the

indifferent uninformed consumer. Let the locations of the indifferent informed/uninformed

consumer when the firm charges the price p, x∗∗i (p) and x∗∗u (p), be solution to this system of

equation, then the firm sets p to maximize its profit: π = p(1− β)((1− η)x∗∗i (p) + ηx∗∗u (p)).

From the first order condition:44

p∗ =

(t((1− η)2 + η β (1− η)

)+ (1− η) vη (1− β)

)θ − t2 (1− η)− 2 η tv

2 θ η (1− β) (1− η)− 2 t (1 + η).

The equilibrium indifferent consumers x∗∗i and x∗∗u can be obtained replacing this expres-

sion into x∗∗i (p) and x∗∗u (p). Equilibrium profits and the surplus enjoyed by informed and

uniformed consumers are respectively:

π = p∗(1− β)((1− η)x∗∗i + ηx∗∗u ),

Si = (1− η)(1− β)

[∫ x∗∗i

0(v + θ((1− η)(1− β)x∗∗i + η(1− β)x∗∗u )− tµ− p∗) dµ

+∫ 1

x∗∗i

(v + θ((1− η)(1− β)(1− x∗∗i ) + β)− t(1− µ)) dµ

],

43This is a sketch of the proof. Full details are available on request.44The second order condition has been verified.

30

Su = η(1− β)

[∫ x∗∗u

0(v + θ((1− η)(1− β)x∗∗i + η(1− β)(x∗∗u ))− tµ− p∗) dµ

],

while the net consumer surplus of those who are mandated to adopt OSS is:

Sm = β

[∫ 1

0

(v + θ((1− η)(1− β)(1− x∗∗i ) + β)− tµ) dµ

].

Total welfare is: W = Si + Su + Sm + π.

31