Cooperative and Noncooperative R&D in Two-Sided Markets

Appendix A: Utility Function and Social Welfare

Following Singh and Vives (1984), the utility of the representative consumer is given by

U(q, m)=∑i=1, 2g=B, S(qig−12(qig)2)−(q1Bq2B+q1Sq2S−αq1Bq1S−αq2Bq2S)+m,

where q=(q1B, q2B, q1S, q2S) is the vector of quantities, and m is a numeraire good produced by a competitive sector.^[18] With this formulation, there is a complementarity only between firm i’s B and S products, not between firm i’s B product and firm j≠i’s S product. The consumer’s net utility is equal to U¯(q, m)=U(q,m)−∑pigqig. The demand system is obtained by solving the four first-order conditions, ∂U¯/qig=0, with i=1,2 and g=B, S.

The consumer surplus is then

CS=U(q,m)−∑i=1, 2g=B, Spigqig−m.

Replacing for the expressions of piB and piS in (1) and (2), we obtain

CS=12[(q1B+q2B)2+(q1S+q2S)2−2α(q1Bq1S+q2Bq2S)].

In the symmetric equilibrium of the quantity subgame, q1B=q2B=q1S=q2S=q, hence, CS=2(2–α)q².

Social welfare, W, is the sum of producer surplus (PS) and consumer surplus (CS), that is,

W=∑i=1, 2∑g=B, Spigqig−C(qiB, qiS, xi, xj)+12[(q1B+q2B)2+(q1S+q2S)2−2α(q1Bq1S+q2Bq2S)].

Appendix B: Second-Order and Positivity Conditions

Under the no-cooperation scenario, the SOC holds iff γ>4(1–α)(2–2α–β)²/(3–2α)²(1–2α)², and equilibrium R&D investments are positive iff γ> 4(1–α)(1+β)(2–2α–β)/(3–2α)²(1–2α)². Under the cooperation scenario, the SOC holds iff γ>4(1–α)(5–8β(1–α)–8α+4α²+β²(5–8α+4α²))/(3–2α)²(1–2α)², and equilibrium R&D investments are positive iff γ>4(1+β)²(1–α)/(3–2α)². Comparing these different thresholds, we find that the SOCs hold and the R&D investments are positive iff γ>max{4(1–α)(5–8β(1–α)–8α+4α²+β²(5–8α+4α²))/(3–2α)²(1–2α)², 4(1+β)²(1–α)/(3–2α)²}, which corresponds to Assumption 1.

Appendix C: Comparative Statics

We determine the variations of R&D investments with respect to the degree of spillovers β and the degree of externalities α.

No-cooperation. First, we have

∂xiNC∂β=−4(1−A)(1−α)(γ(3−2α)2(1−2α)−4(1−α)(2−2α−β)2)(γ(3−2α)2(1−2α)+4β2(1−α)−81−α)2−4β(1−α)(1−2α))2.

Since γ(3–2α)²(1–2α)²>4(1–α)(2–2α–β)² from Assumption 1, and α∈[0,1/2), then γ(3–2α)²(1–2α)>4(1–α)(2–2α–β)², which proves that ∂xiNC/∂β<0. Second, we find that

∂xiNC∂α=2γ(1−A)(3−2α)[(2−2α)(4−2α(3−2α))−β(7(1−2α)+8α2)][γ(3−2α)2(1−2α)−4β(1−α)(1−2α)−8(1−α)2+4β2(1−α)]2.

∂xiNC/∂α has the sign of its numerator, which a decreasing function of β. Since at β=1 the numerator is equal to 2γ(1–A)(3–2α)(1–2α)³>0, ∂xiNC/∂α>0 for all α and β.

Cooperation. As A<1 and α∈[0,1/2), we have

∂xiC∂β=4(1−A)(1−α)(γ(3−2α)2+4(1−α)(1+β)2)((3−2α)2γ−4(1+β)2(1−α))2>0,

and

∂xiC∂α=2(1−A)γ(1+β)(3−2α)(1−2α)((3−2α)2γ−4(1+β)2(1−α))2>0.

Appendix D: Variations of the Direct and Indirect Effects

Let π˜i denotes firm i’s profit gross of R&D costs, that is, π˜i=piBqiB+piSqiS−(A−xi−βxj)(qiB+qiS). The effect of firm i’s R&D investment on its gross profit is given by

dπ˜idxi=∂π˜i∂xi︸direct effect+∂π˜i∂qjB∂qjB∗∂xi︸indirect effectn∘1+∂π˜i∂qjS∂qjS∗∂xi︸indirect effectn∘2,

where qjg∗(x1, x2) designates firm j’s quantity at the equilibrium of the quantity subgame, for good g=B, S.

Appendix E: Proof of Proposition 2

To prove the proposition, we study the sign of ΔW=W^C–W^NC. We find that ΔW has the sign of [2β(1–α)–1]K(α,β,γ), where K(α,β,γ)=–4(1–α)(1+β)²(4(1+β)–3α(3+4β)+α² (4+6β))+(3–2α)²(2–4(2–α)α+(8–2α(9–4α))β)γ. For the following proof, we assume that Assumption 1 holds for all β∈[0,1], which means that γ should be larger than 20/9.

Let K⁽ⁿ⁾=∂ⁿK/∂βⁿ. We find that (i) K⁽³⁾≥0 iff α≥0.42, (ii) K⁽²⁾|_β=₀≥0 iff α≥0.47 and K⁽²⁾|_β=₁≥0 iff α≥0.45, (iii) K⁽¹⁾|_β=₀≥0 for all α and γ>20/9, and K⁽¹⁾|_β=₁≥0 if α≥0.27 or if α<0.27 and γ is sufficiently high (and in particular, larger than 20/9). We study the values of K at the two extremes, β=0 and β=1. We find that (i) K⁽¹⁾|_β=₁>0, and (ii) K|_β=₀>0 if α≤0.21, K|_β=₀<0 if α>0.30, and if α∈[0.21, 0.30] then K|_β=₀>0 if γ is sufficiently high (and in particular, larger than 20/9).

If α≥0.21, since K⁽³⁾≤0, K⁽²⁾ is decreasing. Since K⁽²⁾|_β=₀≤0, we have K⁽²⁾≤0. Therefore, K⁽¹⁾ is decreasing. Since K⁽¹⁾|_β=₁≥0 for high values of γ, we have K⁽¹⁾≥0. This implies that K is increasing. Since K|_β=₀>0 if α≤0.21, K is always positive, and therefore ΔW has the sign of [2β(1–α)–1]. This shows that when the degree of externality is low, social welfare is higher under no cooperation than under cooperation if β≥1/[2(1–α)].

If α>0.47, since K⁽³⁾≥0, K⁽²⁾ is increasing. Since K|_β=₀≥0, K⁽²⁾ is positive, which implies that K⁽¹⁾ is increasing. Since K⁽¹⁾|_β=₀≥0, K⁽¹⁾ is positive, which implies that K is increasing. We have K|_β=₀<0 since α>0.47, and K|_β=₁>0. Therefore, ΔW has the sign of [2β(1–α)–1] if β is close to 1, and the sign of –[2β(1–α)–1] if β is close to 0. This implies that ΔW≥0 if β is close to 1 or close to 0.

Therefore, if α≤0.21, social welfare is higher under no cooperation than under cooperation if and only if (8) holds. If α>0.47, and if β is close to 0, ΔW is also positive.

Appendix F: One-sided R&D investments

In the case of one-sided cost reductions, we find the following equilibrium R&D investments:

x1NC=x2NC=2(1−A)(1−α)(1+2α)(3+2α)2(2−2α−β)(9−4α2)2(1−4α2)γ−2(1+β)[8α4(3+β)+9(2−β)−2α2(23−β)],

and

x1C=x2C=2(1−A)(1−α)(3+2α)2(1+β)(9−4α2)2γ−2(1+β2)(9−8α2).