Introduction The model Solution of the General Model Conclusion

Industry equilibrium with open-source andproprietary firms

by: Gaston Llanes and Ramiro Elijalde

Camilo Pecha

IDEA-BGSE

December 2, 2013

Introduction The model Solution of the General Model Conclusion

Introduction

Open Source (OS): freedom to use, modify and copy sourcecode.

It is evident that there is a coexistence of OS and Proprietaryfirms.

But:

What motivates firms to participate in OS?What are the implications of the coexistence (competition)?Which OS or P firms will produce with higher quality?

Introduction The model Solution of the General Model Conclusion

What is an OS firm?

General Public License and code sharing.

Firms generate profit by selling complementary goods.

Example: IBM invested USD$1 billion in Linux in 2001 andtoday provides support for over 500 software products runningon Linux, and has more than 15,000 Linux-related customersworldwide.

Coexistence examples:

Operating systems: Linux with Mac OS and WindowsWeb browsing : Firefox with Safari or Internet Explores

Introduction The model Solution of the General Model Conclusion

Overview of the model

Game: Two-stage non-cooperative game, n firms and acontinuum of consumers

Firms decide in the first stage to be either OS or P, and in thesecond how much to invest in R+D and price of the good.

IMPORTANT: OS firms share the investment in R+D and Pfirms do not

consumers have vertical and horizontal differentiation (inquality and product respectively)

Introduction The model Solution of the General Model Conclusion

Main findings

Equilibrium with both kinds coexisting: there is anasymmetric market structure with few but large P firms andmany but small OS firms where P firms have a higher quality.

Other results may exist as equilibrium where all firms decideto be OS and there is a distribution of quality.

Introduction The model Solution of the General Model Conclusion

Summary

Important aspects:

There is evidence of industry equilibrium with OS and P firmscoexisting.

Firms sell packages with complementary good.

In the model, decision to be OS is endogenous (in theliterature, these firms are always taken as-is)

Main results

Co-existence may happen as an equilibrium outcome

Forces leading to an asymmetric market structure (few P,many OS)

Complementarities may lead to high quality OS products

Introduction The model Solution of the General Model Conclusion

Technology

Fixed number of firms: n.

Invest on R+D: xi

Fixed cost of investment: F = cxi , and zero marginal cost ofproduction packages

Packages: composed by primary good (OS product) andcomplementary good (P good)

Quality of primary good (OS good)

qi = α ln

(∑i∈OS

xi

)+ (1− α) ln(xi )

α ∈ [0, 1] is the degree of public good R+D investment

Quality of complementary good (P good)

qi = ln(xi )

Introduction The model Solution of the General Model Conclusion

Preferences

Continuum of consumers. Each consumer buys one packagefrom where generates and indirect utility of the form:

vij = qi + y − pi + εij

Vertical differentiation (qi ) and horizontal productdifferentiation (εij)

εij is a taste shock such that

Each consumer has n shocks (one for each good)εij have double exponential distribution (logit model)

Introduction The model Solution of the General Model Conclusion

Demands

Consumer problem: Each consumer observes prices andqualities and then chooses the package that yields the highestindirect utility.

Since the total mass of consumers is 1, so aggregate demandsare equivalent to market shares.

Demand (market share of firm) for good i :

si =exp(δ(qi − pi ))∑

exp(δ(qi − pi ))

as δ increases, the degree of horizontal differentiation amongvarieties decreases.

Introduction The model Solution of the General Model Conclusion

Game and Equilibrium concept

Players: n firms.

Two-stage non-cooperative game:1 Firms decide to become OS or P2 Firms decide invest in R+D and price

SPNE

Symmetric equilibrium in Second Stage:All firms of the same type (from initial stage) play sameequilibrium strategy in the second stage.

Introduction The model Solution of the General Model Conclusion

Solution of the Second Stage I

Recursive solution.nO : the number of firms deciding to be OS (that is given forthe second stage)Second stage problem of the firm i :

maxpi ,xi≥0

πi = sipi − cxi

and from imposing symmetry and FOC: The optimal price:

pi =1

δ(1− si )

And the optimal investment:

xO =1

csO

(1− α nO − 1

nO(1− sO)

), xP =

1

csP

substituting the expressions for xi in qi and pi in the demandswe obtain a system of equations determining si (ni ) fori = O,P

Introduction The model Solution of the General Model Conclusion

Second Stage Equilibrium

Proposition: Second-stage equilibrium exists and is unique.Given nO , market shares solve nOsO + (n − nO)sP = 1 and:

(1−δ) ln

(sOsP

)+

1

1− sO− 1

1− sP= δ ln

(1− α nO − 1

nO(1− sO)

)+αδ ln(nO)

proof i.e. difference in market shares depends on the resolution ofconflict between free-riding and collaborationInterpretation of the RHS:

δ ln

(1− α nO − 1

nO(1− sO)

)︸ ︷︷ ︸

Free-riding:differences in individual investment

+ δα ln(nO)︸ ︷︷ ︸collaboration: xi in OS are multiplied by nO

Introduction The model Solution of the General Model Conclusion

Second stage:Bottom line

Existence an Uniqueness of the equilibrium (sO , sP)

The Market shares (demands) sO and sP can be representedas functions of nO

The trade off between Free-riding and Collaboration is solvedby α and nO

Profits πO and πP can be expressed as functions of thenumber of OS firms (nO)

Introduction The model Solution of the General Model Conclusion

Solution of the First Stage

Profits: (replacing optimal values for prices and investments formsecond stage)

πO(nO) =sO

1− sO

(1

δ− (1− sO) + α

nO − 1

nO

)

πP(nO) =sP

1− sP

(1

δ− (1− sP)

)where si = si (nO)A number nO is an equilibrium if and only if:

πO(nO) ≥ πP(nO − 1), i.e. there is not incentives for OSfirms to deviate and becoming P.

πP(nO) ≥ πO(nO + 1) i.e. there is not incentives for P firmsto deviate and becoming OS.

D’Aspremont et al. (1983) called these conditions internally stableand externally stable coalition conditions

Introduction The model Solution of the General Model Conclusion

Co-existence Equilibrium

Using the function f (nO) = πO(nO)− πP(nO − 1), equilibriumconditions can be restated as f (nO) ≥ 0 and f(nO + 1) ≤ 0.

Figure : Equilibrium number of firms on OS

Introduction The model Solution of the General Model Conclusion

All firms are OS in equilibrium

Figure : Equilibrium number of firms on OS

Introduction The model Solution of the General Model Conclusion

Solution of the First Stage

Proposition: A sub game perfect equilibrium exists. Given n > 3and δ, thresholds 0 < α < α < 1 exists such that:

If α > α, both types of firms coexist and P firms have higherquality and market share than O firms.

If α < α < α, all firms decide to be O, but a P firm wouldhave higher quality and market share.

If α > α, all firms decide to be O, and a P firm would havelower quality and market share.

proof

Introduction The model Solution of the General Model Conclusion

Equilibrium Regions

Figure : Equilibrium Regions

Introduction The model Solution of the General Model Conclusion

Other performed analysis

Welfare: Product quality is suboptimal regardless of thenumber of OS and P firms:

OS firms are subject to free-riding, which leads to asuboptimal investment in R+DP firms do not share their improvements on the primary good,generating an inefficient duplication of effort

Lower differentiation for OS products: nested logit marketshare analysis, however the initial results hold and co-existencestill works.

Introduction The model Solution of the General Model Conclusion

Conclusion

Main Ingredients:

Model of industry equilibrium with OS and P firms.

OS profit from selling a complementary good.

Decision to be OS is endogenous.

Main results:

Co-existence can arise as an equilibrium outcome.

Decision to be OS: optimal business strategy.

Forces leading to an asymmetric market structure.

Complementarities may lead to high quality OS products.

Testable implications.

Introduction The model Solution of the General Model Conclusion

Sketch of the proof I

The idea to proof the existence and uniqueness it is needed toprove two things:

First, only one fixed point of the system of equations exists(only one symmetric equilibrium exists).So computing the first derivative of the equation:

g(sO) = (1− δ) ln(

(n−nO)sO1−nOsO

)− δ ln

(1− α nO−1

(1−sO)nO

)−αδ ln(nO)− n−nO

n−1−nO(1−sO) + 11−sO

that is strictly positive for sOnO ≤ 0, so there exists a unique(sO , sP) that solves the system of equations

Introduction The model Solution of the General Model Conclusion

Sketch of the proof II

Second, the profit function is concave at the equilibrium (thesecond-order conditions for optimality hold).To this end it is enough to find the second derivatives of theprofit functions for OS and P firms and see if those aredefinite negative (using the determinant of the Hessian forwhich it is a sufficient condition to be positive if δ ≤ 1)

back

Introduction The model Solution of the General Model Conclusion

Sketch of the proof I

Existence: For nO = 1 to be an equilibrium, we only needf (2) ≤ 0. Likewise, for nO = n to be an equilibrium, we only needf (n) ≥ 0. To have an equilibrium with both types of firms(1 < nO < n), we need f (nO) ≥ 0 and f (no + 1) ≤ 0 at theequilibrium no. Suppose no equilibrium exists with nO = 1 ornO = n. Then f (2) > 0 and f (n) < 0, so f (nO) goes from positiveto negative at least once when going from nO = 1 to nO = n.Therefore, existence of an equilibrium is guaranteed.

Introduction The model Solution of the General Model Conclusion

Sketch of the proof II

Next, we show f (2) > 0 for any n, α, and δ, which means theequilibrium always has at least two O firms. Let sO be themaximum value of sO(2) for which f (2) ≤ 0. Let w = g(sO)where g(sO) is the condition in Proposition 1. w < 0 impliessO(2) > sO , which means f (2) > 0.The upper bound of w , w is strictly convex in δ and α, whichmeans the maximum is at δ = 0 or δ = 1, and α = 0 or α = 1. Itis straightforward to show that w goes to zero as δ or α go tozero, and also when both δ or α go to 1. Given that w is strictlyincreasing in n, we conclude that w(n, δ, α) is negative for anyfinite n. Therefore, f (2) > 0. back