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Licensed Unlicensed Requires Authentication Published by De Gruyter March 16, 2016

Does a Platform Monopolist Want Competition?

Andras Niedermayer EMAIL logo


We consider a software vendor first selling a monopoly platform and then an application running on this platform. He may face competition by an entrant in the applications market. The platform monopolist can benefit from competition for three reasons. First, his profits from the platform increase. Second, competition serves as a credible commitment to lower prices for applications. Third, higher expected product variety may lead to higher demand for his application. Results carry over to non-software platforms and, partially, to upstream and downstream firms. The model also explains why Microsoft Office is priced significantly higher than Microsoft’s operating system.

JEL: D41; D43; L13; L86

Corresponding author: Andras Niedermayer, Department of Economics, University of Mannheim, L7, 3-5, D-68131 Mannheim, Germany, e-mail:


I thank the editor, Julian Wright, an anonymous referee and Philipp Ackermann, Simon Anderson, Werner Boente, Ingo Borchert, Stefan Buehler, Alain Egli, Winand Emons, Roland Hodler, Simon Loertscher, Gerd Muehlheusser, Daniel Niedermayer, Ferenc Niedermayer, Armin Schmutzler, Dezsö Szalay, and Lucy White for very helpful comments. I further thank participants of seminars at the University of Bern, SSES 2006 in Lugano, IIOC 2006 in Boston, Swiss IO Day 2006 in Bern, EARIE 2006 in Amsterdam, and YSEM 2006 in Bern for valuable discussions. The author gratefully acknowledges grants by the Swiss National Bank, the Verein für Socialpolitik, grant PBBE1-121057 from the Swiss National Science Foundation and support from the Deutsche Forschungsgemeinschaft through SFB-TR 15.


A Examples of Monopolists Inducing Competition

Examples of platform owning companies which also sell an application running on their platform are provided in Table 1. Note that these firms have encouraged competition (or at least not prevented it) in their applications market in one form or another and that they make a significant part (or even most) of their profits with their application(s) and not only with their platform. Intel as an example in Table 1 is taken from Besen and Farrell (1994). The authors also give examples of how a monopolist may encourage usage of its standard (or competition on its platform): “Concessions [to encourage adoption of the standard] include ... actions that make it more attractive for the other firm to use [the monopolist’s technology]: low-cost licensing, hybrid standards, commitment to joint future development, shifting standards development to third parties, and promising timely information to rivals.” Microsoft Windows and the Nintendo Entertainment System are described among other examples in detail in Evans et al. (2004). Evans et al. (2004) write that “Microsoft ... realized that ... it made sense to make it as attractive as possible to write software for their platform.” They further write that “Nintendo [was the first console maker who] actively pursued licensing agreements with game publishers” and that Nintendo relied “on revenue from games produced in-house along with royalties from games sold by independent developers” and did not make profits with the console itself.

Table 1:

Examples of Platform Owners Who are Also Active on One Side of the Market.

CompanyPlatformOwn applicationCompeting application(s)
MicrosoftWindowsExcel (more generally MS Office)Lotus 1-2-3, OpenOffice Calc
AdobePDF file formatAcrobat StandardPDF Writer, PDF Creator
GooglePage with search resultsPaid ads sold by GoogleLinks to third-party web pages with ads
Intel Corp.Intel compatible processorsIntel processorsAMD processors
NintendoNintendo Entertainment Systems57% of games (e.g. Super Mario)e.g. Dragon Warrior, Final Fantasy
Non-software examples
CompanyPlatformOwn shopCompeting shop(s)
MigrosbGlatt Zentrum (shopping mall in Zurich, Switzerland)Hotelplan (travel agency)Kuoni, Imholz/TUI
CoopbWankdorf Center (shopping mall in Bern, Switzerland)Coop RestaurantSegafredo

aIBM is not the owner of Linux. However, they have invested significant amounts (estimated to be more than $1bn) in Linux and employ over 300 Linux Kernel developers. IBM could have just as well promoted one of its proprietary operating systems (such as OS/2) which would have given them better chances to exclude competing application vendors. IBM claims to have recouped investments in Linux with increased application and hardware sales.

bMigros and Coop are major retailers in Switzerland.

Adobe’s PDF file format is an example as well, with the file format as a platform and software for creating files as applications. Adobe first intended the PDF file format to be a proprietary file format. But at the beginning of the 1980s they decided to open the file format to competitors. This move helped PDF to become one of the leading formats for electronic documents. This example fits our modification of a zero price platform well: Adobe does not charge royalties for the file format, however, they make money with software for the creation of PDF files.[35]

A further example where the model can be applied are research areas. One can consider a strain of research literature as a platform, articles in this strain as applications and readers of articles as consumers. Getting acquainted with a research area incurs investment costs and readers do not know ex ante whether the articles are worth the effort. Therefore, if there are more articles in a certain area, they are more willing to make this investment. Hence, more articles in a research area have two opposing effects on people already working in it: there are more readers of this strain of literature, but there is tougher competition for readers, as well. Either of the two effects might be stronger. The same argument applies for the choice of a language of publication as for the choice of a research area.

An example loosely related to our model which also shows similar effects (a firm wanting competitors to enter the market) are chain stores with a franchising system, e.g. McDonald’s, as described in Loertscher and Schneider (2011). Consider consumers who move to another area with a certain probability and who face switching costs if they go to a different chain store in the future. Such consumers are more willing to buy a franchisee’s products if there are more other franchisees of the same franchisor elsewhere.

An example of upstream/downstream firms for which this model can be applied as well is the case of AMAG Automobil- und Motoren AG. AMAG is the exclusive importer of Porsche in Switzerland and also has several branches selling Porsches directly to customers. However, they also sell Porsches to independent garages.

B A Simple Setup with Complementary Products

To illustrate why a platform owner would set a lower price for its complementary application, consider a representative consumer consuming xp units of the platform, xA units of application A, and xB units of application B. Assume that A and B are perfect substitutes and that the platform and an application are perfect complements. The consumer’s preferences can thus be represented with U(xp, xA, xB)=min{xp, sA+xB}.

If application B is not available, it is clear that only the sum of the prices of the platform pP and the application A pA matter to the consumer’s consumption choice and the firm’s profit maximization problem. If the consumer spends his whole income m on the system, his consumption will be xP=xA=m/(pP+pA) and the monopolist’s revenue (pP+pA)xP (which is of course the whole income m).

Now imagine that B enters the market. The consumer will buy the same amount of the platform and the cheaper of the two applications, i.e. xP=xi=m/(pP+min{pA, pB}) with i being the cheaper application. B will always try to undercut a positive price pA because of the usual reasoning for Bertrand competition. A even has a further incentive to set pA below pB as it also increases his profits made with the platform. Therefore, in equilibrium pA=pB=0 and pP is some positive price.

To sum up, whenever a substitute is present for A’s application, he will set the application’s price to 0 and only make profits with his platform.

B.1 Quality Differences

This simple setup can be altered to account for quality differences between the applications. Consider a consumer with the quasilinear utility function


xO representing all other goods at price pO=1. cAxA and cBxB represent disutility from say having to learn to use an application. The lower c(·), the higher the quality of the application. v is a concave function such that equilibrium demand and prices are positive and finite.[36]

The consumer will pick application i with the lower ci+pi. The Lagrangian of the consumer’s maximization problem is L=v(xS)–cixS+xOλ(pPxS+pixS+xOm), where xS=xP=xi is the number of units of the platform-application system consumed. The first order condition with respect to xO implies λ=1. Hence, ∂L/∂xS=v′(xS)–cipPpi=0 and the consumer’s demand for the system xS is[37]


If application B is not available or the consumer prefers application A (i.e. cA+pA<cB+pB), A’s profit is (pP+pA)xS under the assumption of zero production costs. Here, again, only the sum p=pP+pA matters. A chooses p*=arg maxppD(p+cA) and his profit is p*D(p*+cA).

If the consumer chooses application B (i.e. cB+pB<cA+pA), Nash equilibrium prices are given by

pP=argmaxpPpPD(pP+pB+cB),pB=argmaxpBpBD(pP+pB+cB)   s.t. pBcAcB+pA,

for some pA. Clearly the sum of equilibrium prices will be weakly higher than in the monopoly case, pP+pBp, but demand D(pB+pB+cB) and A’s profits pPD may still be higher if cB is sufficiently low (see example below). If A’s profits are higher when B’s application is bought by consumers, he will encourage entry by B and not sell his own application. If A’s profits are lower when B’s application is bought, he will try to prevent B from selling by setting up technical and legal obstacles and by setting a low price pA for his application.

Example. Take for simplicity a function v which leads to a linear elasticity of demand, v(xS)=γ(δ+1–lnxS)xS. Demand is hence D=(v′)–1=exp(δ–(pP+pi+ci)/γ). If A sells both the platform and the application, he sets p*=arg maxppD(p+cA)=γ and his profits are

(B.1)ΠM=γeδ1+cA/γ. (B.1)

If B sells the application, Nash equilibrium prices are pP=γ and pB=γ granted that B’s constraint pBcAcB+pA is not binding. A’s profit is

(B.2)ΠC=γeδ2+cB/γ. (B.2)

If cAcB>γ the constraint on pB is not binding whatever pA, also under the same condition A’s competition profits (B.2) are higher than his monopoly profits (B.1). For cAcB∈[0, γ] B selling his application at price pB=γ would be disadvantageous for A, therefore, he sets pA=0 which results in either pB=cAcB or in A taking over the application market. In either case, A’s profit is ΠM. For cAcB<0 A does not want B to sell his application B and can prevent him from doing so by setting pA<cBcA.

To sum up, whenever B’s entry in the application market is advantageous for A, A will let B take over the application market and make money with his platform only. Therefore, a simple setup is not sufficient to explain why a monopolist would want to encourage entry in the application market and make profits with his application at the same time.

C Alternative Cases of Monopoly

If B does not enter, A is a monopolist at stage 2. Here two possibilities exist: if sA is sufficiently large (sA≥2t) A will serve all consumers (full market coverage, see Figure 8A), otherwise (sA<2t) A will charge such a high price that some of the consumers will not buy the application (partial market coverage, Figure 8B).

Figure 8: Cases of Monopolistic Pricing by A. The Vertical Axis Denotes Excess Utility vA–v0 Derived from the Usage of Application A.(A) Full Market Coverage. (B) Partial Market Coverage.
Figure 8:

Cases of Monopolistic Pricing by A. The Vertical Axis Denotes Excess Utility vAv0 Derived from the Usage of Application A.

(A) Full Market Coverage. (B) Partial Market Coverage.

We will derive the condition that separates the two cases.

Firm A’s profits from application sales are πA=pAx^ where x^ denotes the location of the consumer furthest away from A who is still willing to buy the application. If only part of the consumers buy the application x^ is the indifferent consumer with x^ satisfying sApAtx^=0 and, therefore, x^=(sApA)/t. If all consumers are willing to buy the platform, i.e. even the consumer at location 1 has a non-negative utility from buying the platform sApA–t≥0 has to be satisfied and x^ is equal to 1.

Formally we get

x^={(sApA)/tif (sApA)/t<1,1otherwise.

Proposition 1 in the main text derives the separating condition and shows the equilibrium for the full coverage case. We provide a proof of Proposition 1 in the following.

Proof of Proposition 1. Substituting pA=pA and x=1 into excess utility vAv0=sApAtx yields vAv0=0. Therefore, for pA=sAt the consumer at location x=1 is just indifferent between buying and not buying. Demand is hence 1 and profits are πA=pA=sAt. It does not pay off to choose a lower price pAl<pA because demand cannot be larger than 1 and profits are hence πAl=pAl<pA=πA. It does not pay off either to choose a higher price pAh>pA. For a higher price demand would be x^=(sApA)/t which is less than 1. Profits would be πAh=pAh(sApAh)/t and the derivative of the profit function πAh/pAh=sA/2pA. At pAh=pA (and hence at x^=1) the derivative is tsA/2. For sA≥2t the derivative of the profit function is non-positive at pAh=pA and decreasing in pAh, therefore, πAhπA and the firm is not willing to increase its price.          □

For the case where sA≤2t the monopolist sells only to a part of the consumers.

His profit maximization problem is


Solving the first order condition for pA yields pA=sA/2. The location of the marginal consumer and profits are hence x^*=sA/2t and πA=sA2/4t. Consumer surplus is


D Alternative Cases of Competition

Five cases can be distinguished in a fixed location Hotelling setup: 1) an “inner equilibrium” (sA+sB>3t and –3t<sAsB<3t, see Figure 9A), 2) market domination by A (sA+sB>3t and sAsB≥3t, Figure 9B), 3) market domination by B (sA+sB>3t and sAsB≤–3t, Figure 9C), 4) two local monopolies (sA+sB≤2t, Figure 9D) and 5) a “limiting case” where prices are too low for a local monopoly, but too high for competition (2t<sA+sB≤3t, Figure 9E).

Figure 9: Different Cases in a Hotelling Setup. The Vertical Axis on the Left Denotes the Excess Utility vA–v0 Derived from the Usage of Application A, the Vertical Axis on the Right Denotes the Excess Utility vB–v0 from B.(A) “Inner Equilibrium”. (B) A Captures Whole Market (x˜=1).$(\tilde x = 1).$ (C) B Captures Whole Market (x˜=0).$(\tilde x = 0).$ (D) Local Monopolies. (E) “Limiting Case” (x˜=x˜A=x˜B).$(\tilde x = {\tilde x_A} = {\tilde x_B}).$
Figure 9:

Different Cases in a Hotelling Setup. The Vertical Axis on the Left Denotes the Excess Utility vAv0 Derived from the Usage of Application A, the Vertical Axis on the Right Denotes the Excess Utility vBv0 from B.

(A) “Inner Equilibrium”. (B) A Captures Whole Market (x˜=1). (C) B Captures Whole Market (x˜=0). (D) Local Monopolies. (E) “Limiting Case” (x˜=x˜A=x˜B).

We will derive these conditions and the equilibria arising in the different cases after introducing some notation.

The consumer indifferent between applications A and B will be denoted with x˜ satisfying sApAtx˜=sBpBt(1x˜), the consumer indifferent between buying application A and not buying any application with x˜A satisfying sApAtx˜A=0, and the consumer indifferent between B and not buying with x˜B satisfying sBpBt(1x˜B)=0. Solving for x˜,x˜A, and x˜B yields

(D.1)x˜=12+12t(sAsB+pBpA), (D.1)
(D.2)x˜A=1t(sApA), (D.2)
(D.3)x˜B=11t(sBpB). (D.3)

We will call the demand for application A xA and the demand for application B (1–xB) where

(D.4)xA={0if x˜<0,min{x˜,1,x˜A}else, (D.4)


(D.5)xB={1if x˜>1,max{x˜,0,x˜B}else. (D.5)

The five cases can be formally defined as follows:

  • “Inner Equilibrium”: x˜B<x˜A and 0<x˜<1

  • Domination by A: x˜1

  • Domination by B: x˜0

  • Local Monopolies: x˜A<x˜B

  • “Limiting Case”: x˜=x˜A=x˜B

The following propositions state the conditions for the cases and the resulting equilibria. As in the main section, we will use Δ as a shorthand for sAsB.

Proposition 5If sA+sB>3t and –3t<Δ<3t there is an “inner equilibrium” (x˜B<x˜A and 0<x˜<1) with equilibrium prices pA=t+Δ/3 and pB=tΔ/3.

Proof. Substituting pA and pB into x˜A and x˜B yields

x˜A=2sA+sB3t1,   x˜B=2sB+sA3t+2.

Substituting this into the condition x˜B<x˜A and regrouping yields 3t<sA+sB which is fulfilled by assumption.

Substituting pA and pB into x˜ we get x˜=1/2+Δ/6t. The condition 0<x˜<1 can be rewritten as –3t<Δ<3t which is again fulfilled by assumption.

Because both x˜B<x˜A (and thus x˜B<x˜<x˜A) and 0<x˜<1 hold we can write the demand functions specified in (D.4) and (D.5) as xA=x˜ and 1xB=1x˜. The Nash equilibrium is hence


Solving the first order conditions of the two maximization problems for pA and pB yields pA=t+Δ/3 and pB=tΔ/3.          □

Proposition 6If sA+sB>3t and Δ≥3t A will capture the whole market(x˜1)and equilibrium prices arepA=sAsBtandpB=0.

Proof. Substituting pA and pB into x˜ yields


B has no incentive to deviate from pB=0: with a negative price her profits would be non-positive, with a higher price her demand would remain zero.

A has no incentive to deviate either. With a lower price his demand would still be 1, therefore, his profits would decrease.

The reason why he would not set a higher price is the following. At pA=pA the derivative of the profit function πA=pAx˜ is


The derivative is non-positive at pA for sAsB≥3t and linearly decreasing in pA. Therefore, A has no interest in increasing the price.□

Proposition 7If sA+sB>3t and Δ≤–3t B will capture the whole market (x˜0) and equilibrium prices are pA=0 and pB=sBsAt.

Proof. By analogy to Proposition 6.          □

Proposition 8If sA+sB<2t there are local monopolies (x˜A<x˜B) and equilibrium prices are pA=sA/2 and pB=sB/2.

Proof. Substituting pA and pB into x˜A and x˜B yields x˜A=sA/2t and x˜B=1sB/2t. Substituting this into x˜A<x˜B gives sA/2t<1–sB/2t which is equivalent to sA+sB<2t and hence fulfilled by assumption.

x˜ has to be between x˜A and x˜B, therefore, we can write demand as xA=x˜A and 1xB=1x˜B. The two local monopolists do not compete with each other, hence the two firms choose profit maximizing prices independently and we get


by solving the first-order conditions.          □

When neither of the aforementioned cases occurs (2tsA+sB≤3t), we have the “limiting case” with x˜=x˜A=x˜B.

E Monopolist with Two Applications

One can consider the case where the platform owner owns both application A and application B, either because he has developed application B himself or because he acquired firm B. Let fAB be firm A’s fixed costs of developing application B[38] or the price he has to pay to acquire the competitor.

Similarly to Section 3 we assume that sA+sB>2t.[39] (In the case sA+sB<2t the two-application monopolist would not cover the whole market and we have the same case as two independent local monopolies as described in Appendix C.)

At stage 2 the monopolist sets prices such that consumer x˜ who is indifferent between buying an application and not buying, i.e. vA(x˜)=vB(x˜)=v0 or


We can solve this double equation for both x˜ and pB. The profit maximization problem becomes maxpA,pB{pAx˜+pB(1x˜)} or


Solving the first order conditions gives us


The condition 0<x˜<1 is fulfilled if –2t<Δ<2t with Δ≡sAsB as previously. For Δ outside of this range the monopolist sells only one of his applications. For stage 2 profits and expected consumer surplus we get


Stage 1 profits are the same as in Eq. (11) except that now we have πA+πB instead of πA

(E.1)Πtwo=α4(s+EU+πA+πB)2=α4[s+t4+3Δ216t+sA+sB2]2. (E.1)

E.1 Comparison with Single Application Monopolist

Now we can compare the single application monopolist’s profits with the profits of the two application monopolist. Developing a second application (or acquiring the competitor) incurs fixed costs fAB, therefore we have to compare Π*twofAB with Π*M from Eq. (11). Developing a second application pays off for the monopolist if Π*twofAB*M. It can be (trivially) seen that for fAB sufficiently large, it does not pay off to develop a second application. It can further be shown that if development costs for the second application are zero, it always pays off to develop it (i.e. Π*two–Π*M, see Proposition 9).

Proposition 9 Π*two–Π*M

Proof. Substituting (E.1) and (11) into Π*two–Π*M and regrouping yields 3Δ2–8tΔ+20t2>0. Because the coefficient of Δ2 is positive and the polynomial in Δ has no real roots, the equation is always fulfilled.          □

E.2 Comparison with Competition

Having a competitor compared to developing both products oneself pays off if Π*twofAB*C with Π*C taken from Eq. (13). Again, for fAB sufficiently large, it does not pay off to develop the second application. And again, it can be shown that for fAB=0 it pays off to develop a second application oneself instead of letting the competitor develop it (see Proposition 10).

Proposition 10 Π*two*C

Proof. Substituting Eq. (E.1) and (13) into Π*two*C and regrouping yields 5Δ2–16tΔ+48t2>0. Again, this polynomial in Δ has no real roots and therefore the left hand side is always positive.          □

F Model with Different Distribution of Consumers

Alternatively to the results Subsection 6.2 we can consider a different distribution of consumer preferences in order to get a purely analytical solution: consumers are homogeneous with respect to their preferences for the platform and all have the parameter value y1 as depicted in Figure 10.

Figure 10: Consumers with Homogeneous Preferences y=y1 Over the Platform.
Figure 10:

Consumers with Homogeneous Preferences y=y1 Over the Platform.

We can describe the density of consumers with the Dirac delta function δ(·) used in physics:

ρ{x,y)=(δ(yy1)for 0x1,0otherwise.

The number of consumers between 0 and y˜ is thus

N=0y˜01ρ(x,y)dxdy={1if y˜y1,0otherwise,

i.e. either all consumers buy the platform or none. We will first look at the monopoly case in this setup and then at the competition case. We will show that it is possible that a monopolist cannot sell his platform even if he can commit to the application price at stage 1. Then we shown that in such a situation competition can be a remedy.

F.1 Monopoly

As in Section 3 we assume full market coverage, i.e. the monopolist sets the application price such that consumers with all values of x are willing to buy the application. However, contrary to the previous sections, the outermost consumer (x=1) is not necessarily set indifferent between buying and not buying (see Figure 11), because the monopolist may be willing to commit to a lower pA at stage 1 to convince consumers to buy the platform.

Figure 11: Full Coverage with Price Commitment at Stage 1. The Shaded Area Below the Curve Denotes Consumer Surplus.
Figure 11:

Full Coverage with Price Commitment at Stage 1. The Shaded Area Below the Curve Denotes Consumer Surplus.

For stage 2 profits and expected consumer surplus we get πA=pA and EU=sApAt/2.

The condition for full market coverage at stage 1 is

(F.1)sApAt. (F.1)

At stage 1, consumers are willing to buy the platform if their y is not above


Because all consumers have y=y1, the monopolist has to commit to a price pA at stage 1 such that

(F.2)y˜y1 (F.2)

to ensure that consumers are willing to buy his platform.

The profit maximization problem of the monopolist consists of setting pA as high as possible such that conditions (F.1) and (F.2) are still satisfied. We take the case where condition (F.2) is stronger than condition (F.1) and the monopolist sets pA such that (F.2) is just binding:


For the equilibrium application price we get


and for overall profits


because πA=pA and N*=1.

Now let us consider the case where

(F.3)y1>s+sAt2. (F.3)

In this case the firm would have to set a negative price PA for the application to convince consumers to buy his platform. Hence, in this case it is not possible for the monopolist to get positive profits.

F.2 Competition

If B enters the market, both firms commit to application prices at stage 1. They face the same problem as at stage 2 in the previous sections with the additional constraint that consumers should be willing to buy the platform:

(F.4)y˜y1 (F.4)

where y˜=s+EU is the maximal distance at which consumers are still willing to buy the platform.

We consider the case where (F.4) is non-binding. In this case we can use the results obtained in Subsection 4.1, the only difference is that prices are set at stage 1 and not at stage 2. Equilibrium stage 2 profits and expected consumer surplus are given in Eqs. (16), (17), and (19).

Firm A’s profits are Π=πAN=πA because N*=1.

F.3 Comparison of Profits

In the case where the monopolist cannot achieve positive profits, but with competition profits are strictly positive, firm A is (trivially) better off with competition. This case occurs for parameter values which satisfy both conditions (F.3) and (F.4). Proposition 11 states when both conditions can be satisfied simultaneously.

Proposition 11For Δ(3t,(963)t) conditions (F.3) and (F.4) can both be satisfied at once if neither firm dominates the market.

Proof. Substituting (19) into (F.4) gives


Combining this with (F.3) yields


The range of y1 which allows for both conditions to be satisfied is non-empty if the lower bound of y1 given in the previous equation is less than its upper bound. This is satisfied if Δ2–18tΔ–27t2>0. The roots of this polynomial in Δ are


The polynomial is positive for values of Δ not between the roots. Combining this with the assumption that neither firm dominates the market (–3t<Δ<3t, see Eq. (13)) we get

3t<Δ<(963)t.          □

G Zero Price Platform and Possibility of Price Commitment: Derivation of Profits

In the following, we derive profits for a setup with a zero price platform and the platform owner’s possibility to commit to prices.

G.1 Monopoly

For the monopoly situation we look at two cases: full and partial coverage. In the full coverage case even the outermost consumer will buy the application at stage 2 (pAsAt). Because the monopolist sets pA already at stage 1, he may set a lower price than the price which sets the outermost consumer indifferent, so that more consumers are willing to buy the platform at stage 1. Monopoly profits are πA=pA per consumer unit, consumers’ expected utility for stage 2 is again the integral over x, and overall profits for full coverage are Π=πAN=pAα(s+sApAt/2). The profit maximizing price is pAfull=argmaxpAΠ=s+sA/2t/4 which satisfies the condition for full coverage (pAsAt) if ssA/2–t/4. Substituting pAfull into πA gives us the maximal profits in the case of full coverage Πfull.

In the partial coverage case the monopolist does not sell to all consumers at stage 2. Profits per consumer unit are πA=pAx^=pA(sApA)/t with x^ being the indifferent consumer. Expected utility is the integral between 0 and x^. Profits are


where N=α(s+EU). The first order condition (∂(πAN)/∂pA=0) of the profit maximization problem is a fifth degree polynomial in pA and gives us five solution candidates. We check for different parameter values whether the solution candidates satisfy the following conditions: price is a nonnegative real number, second order condition, there is an indifferent consumer (0x^1). For all parameter ranges considered this procedure gives us a unique solution. Substituting the optimal price into the profit function gives us the partial coverage profit Πpartial.

The monopolist chooses full or partial coverage depending on where profits are higher.

G.2 Competition

Stage 2 of the competition case is the same as in Subsection 4.1 with the difference that only the choice of consumers has to be considered, because the firms have already committed to a price at stage 1. At stage 1 firms set prices pA and pB taking into account that they influence both platform choice at stage 1 and application choice at stage 2. The Nash equilibrium is thus


We find the Nash equilibria by solving the first order conditions. Then we check for different parameter values whether the obtained solutions candidates (pA,pB) fulfill the following conditions: prices are nonnegative real numbers, there is an indifferent consumer (0x˜(pA,pB)1), the indifferent consumer has a positive excess utility (vA(x˜(pA,pB))v0>0), second order conditions for A and B. For all parameter ranges considered there was no multiplicity of equilibria. However, there were parameters for which no inner equilibrium (i.e. both firms coexist and all consumers with a platform buy an application) was found. In these areas either one of the two firms dominates the market or there are local monopolies. We only consider the inner equilibrium cases and substitute equilibrium prices into A’s profits which gives us Πcomp.


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Published Online: 2016-3-16
Published in Print: 2015-3-1

©2015 by De Gruyter

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