Strategic Compatibility Choice, Network Alliance, and Welfare

Abstract

Based on a simple model of compatibility choice under differentiated Cournot duopoly with network externalities, we consider how the levels of a network externality and product substitutability affect the choice of compatibility. In particular, if the level of network externality is larger than that of product substitutability, there are multiple equilibria involving imperfect and perfect compatibility. Furthermore, we demonstrate the conditions for constructing such a network alliance so that firms provide perfectly compatible products. The network alliance is stable and socially optimal.

Appendix: The case of consumers ex ante expectations

Taking eqs. (1) and (2), the profit function of firm i is given by:

$$ {\pi}_i=\left\{A-{q}_i-\gamma {q}_j+N\left({S}_i^e\right)\right\}{q}_i,\kern0.5em i,j=1,2,\kern0.5em i\ne j. $$

(12)

The FOC of profit-maximization of firm i is:

$$ \frac{\partial {\pi}_i}{\partial {q}_i}={p}_i-{q}_i=A-2{q}_i-\gamma {q}_j+N\left({S}_i^e\right)=0,\kern0.5em i,j=1,2,\kern0.5em i\ne j. $$

(13)

At the point of a fulfilled expectation, i.e., when ${q}_i^e={q}_i$ and ${q}_j^e={q}_j,$ in view of eqs. (2) and (13), we obtain the following:

$$ A-\left(2-n\right){q}_i-\left(\gamma -n{\phi}_j\right){q}_j=0,\kern0.5em i,j=1,2,\kern0.5em i\ne j, $$

(14)

Thus, we derive the following fulfilled expectation Cournot equilibrium:

$$ {q}_i^{\ast \ast }=\frac{\left\{2-n-\left(\gamma -n{\phi}_j\right)\right\}A}{\varDelta },\kern0.5em i,j=1,2,\kern0.5em i\ne j, $$

(15)

where Δ ≡ (2 − n)² − (γ − nϕ ₁)(γ − nϕ ₂) > 0. Using eq. (13), the profit of firm i can be expressed as: ${\pi}_i^{\ast \ast }={\left({q}_i^{\ast \ast}\right)}^2,$ i = 1, 2..

Given eq. (15), the effects of an increase in the level of compatibility of the firms on the equilibrium output are given by:

$$ \frac{\partial {q}_i^{\ast \ast }}{\partial {\phi}_i}=\frac{n\left(n{\phi}_j-\gamma \right)}{\varDelta }{q}_i^{\ast \ast }>\left(<\right)0\iff n{\phi}_j>\left(<\right)\gamma, $$

(16)

$$ \frac{\partial {q}_i^{\ast \ast }}{\partial {\phi}_j}=\frac{n\left(2-n\right)}{\varDelta }{q}_j^{\ast \ast }>0,\kern0.5em i,j=1,2,\kern0.5em i\ne j. $$

(17)

We proceed to the game of compatibility choice. Based on eqs. (16) and (17), we derive the following:

$$ \frac{\partial {\pi}_i^{\ast \ast }}{\phi {\phi}_i}=\frac{2n\left(n{\phi}_j-\gamma \right)}{\varDelta }{\left({q}_i^{\ast \ast}\right)}^2>\left(<\right)0\iff n{\phi}_j>\left(<\right)\gamma, $$

(18)

$$ \frac{\partial {\pi}_i^{\ast \ast }}{\partial {\phi}_j}=\frac{2n\left(2-n\right)}{\varDelta }{q}_i^{\ast \ast }{q}_j^{\ast \ast }>0, $$

(19)

where ϕ _j , ϕ _j  ∈ [0, 1], i, j = 1, 2, i ≠ j.

Thus, as shown in the text, we derive the same result as in Proposition 1 in the text. Therefore, by the same procedure as in the text, we can derive the same results as in Propositions 2 and 3.