Disclosure of R&D Knowledge with Cross-Ownership in a Mixed Duopoly

Ting Zhang; Zhaoxin Qi; Dongdong Li

doi:10.1515/bejeap-2024-0215

Article

Disclosure of R&D Knowledge with Cross-Ownership in a Mixed Duopoly

Ting Zhang , Zhaoxin Qi and Dongdong Li

Published/Copyright: July 7, 2025

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From the journal The B.E. Journal of Economic Analysis & Policy

Abstract

This paper investigates the disclosure of R&D knowledge with cross-ownership in a mixed duopoly, where a semi-public ownership firm competes with a private firm. The results show that regardless of the degree of cross-ownership, both semi-public and private firms fully disclose their information. Furthermore, we find that privatization maintains the same disclosure level but reduces social welfare. Finally, we extend the model to unilateral ownership, the presence of foreign-owned firms and strategic R&D investment. We find that under unilateral shareholding, when the semi-public firm holds shares of the private firm, the semi-public firm fully discloses its R&D knowledge, while the private firm does not disclose its R&D knowledge. However, when the private firm holds shares of the semi-public firm, both semi-public and private firms fully disclose their information. The previous result still holds if the private firm is foreign-owned and strategic R&D investment is incorporated.

Keywords: R&D knowledge disclosure; cross-ownership; mixed duopoly

JEL Classification: O30; L13; L32

Corresponding author: Dongdong Li, Northwestern Polytechnical University, Xi’an, China, E-mail: lidd@nwpu.edu.cn

Appendix A

This appendix presents supplementary explanations, the solution of the equilibrium results, and the related assumptions and constraints in the main text.

Section 2: Analysis of Similarities and Differences with d’Aspremont and Jacquemin (1988)

To better highlight the similarities and differences with d’Aspremont and Jacquemin (1988), we analyze the model setup and the transmission mechanisms of information disclosure. Similar to d’Aspremont and Jacquemin (1988), our model adopts a staged game structure for R&D investment (R&D investment → output competition) and emphasizes the impact of R&D spillover effects on market equilibrium. However, unlike the exogeneity of the spillover rate in d’Aspremont and Jacquemin (1988), this paper internalizes the disclosure of R&D knowledge into the firms’ strategic choices through cross-ownership. Furthermore, the dual objectives of the semi-public firm (as shown in Eq. (3)) and the profit-sharing mechanism of the private firm (as shown in Eq. (4)) jointly drive the full-disclosure equilibrium, which distinguishes this from purely private or public markets. Additionally, to better clarify the results of the paper and their economic intuition, we further emphasize the transmission mechanisms of information disclosure in private duopolies, mixed duopolies, and in a duopoly with a semi-public firm competing with a private firm, as illustrated in Table 1.

Table 1:

Analysis of differences under different market structures.

Market structure	Information disclosure equilibrium	Core of the transmission mechanism
Private duopoly (no Cross-Ownership)	β₁ = β₂ = 0	Profit protection motivation dominates
Private duopoly (cross-ownership)	β₁ = β₂ = 1	Profit sharing internalizes spillover benefits
Mixed duopoly (public + private, No Cross-Ownership)	β₁ = 1, β₂ = 0	Public firm drives industry efficiency
Semi-public + private (cross-ownership)	β₁ = β₂ = 1	Cross-ownership synergizes the internalization of social and profit objectives

Specifically, in a private duopoly, firms naturally face a prisoner’s dilemma regarding information secrecy. The classic study by d’Aspremont and Jacquemin (1988) shows that in a private duopoly, exogenous technological spillovers cannot be internalized through non-cooperative mechanisms, leading to insufficient R&D investment. In contrast, this paper internalizes the disclosure level β as a strategic choice for firms through cross-ownership: in a private duopoly, cross-ownership achieves full disclosure through a profit-sharing mechanism (Lemma 4), which has a similar effect to their ‘cooperative R&D’ equilibrium, but without the need for an explicit cooperation agreement. When firm i holds shares of firm j, the disclosure that reduces firm j’s costs generates net profits through profit-sharing, resulting in Pareto-optimal information sharing β = 1. In a mixed duopoly (Bárcena-Ruiz & Garzón 2020), the public firm, acting as a social welfare agent, reduces the private competitor’s costs by fully disclosing (β = 1), expanding industry output to correct market failure. On the other hand, the private firm adheres to non-disclosure (β = 0) to maintain its cost advantage, resulting in an asymmetric equilibrium. This contradiction reveals the inherent conflict between public and private objectives: the public firm’s disclosure strategy essentially weakens its market power, compensating for producer surplus losses with consumer surplus gains. In our model, when a semi-public (partially privatized) firm and a private firm cross-hold shares, the ownership structure creates a “hybrid effect” in the objective functions – the semi-public firm incorporates private profits into its objective (Eq. (3)), and its disclosure behavior serves both social efficiency and its own profit. The private firm, by holding shares in the public firm, internalizes part of the social welfare weight (Lemma 3), making information disclosure the dominant strategy for improving industry efficiency. This dual penetration blurs the traditional boundary between public and private sectors, making full disclosure (β = 1) the optimal solution for cross-ownership collaboration.

Section 3: Mixed Duopoly

First-order and second-order conditions for the equilibrium outputs:

The following first-order conditions emerge:

∂ O 1 M ∂ q 1 M = 4 α − 1 θ − 3 q 1 + α θ − 1 − 1 q 2 − α θ − 1 w + x 1 + x 2 β 2 ∂ O 2 M ∂ q 2 M = α − 1 − α θ q 1 − α − 1 w − 4 q 2 + x 2 + x 1 β 1 .

It is easy to verify that second-order conditions are satisfied.

The expressions represented by w and H 1 − 4 are as follows:

w = a − c H 1 = 11 + 4 θ + α α + α θ 14 + θ − 4 3 + 5 θ H 2 = α θ − 1 H 3 = α − 1 4 α θ − 3 − θ H 4 = 2 + θ + α θ α 3 + θ − 5 − 2 .

First-order and second-order conditions for the equilibrium R&D levels:

The following first-order conditions emerge:

∂ O 1 M ∂ x 1 M = ( H 1 2 ⁢ H 2 ⁢ x 1 − α − 1 2 ⁢ α + H 2 2 ⁢ 4 α − 1 ⁢ θ − 3 ⁢ w + x 1 + x 2 + 2 H 1 ( 1 + α − θ ) H 5 ⁢ w + x 1 + x 2 − 3 + 4 α − 3 θ ⁢ H 5 2 ⁢ w + x 1 + x 2 ) / H 1 2 ∂ O 2 M ∂ x 2 M = − ( ( 4 1 − α ⁢ α θ ⁡ α + 3 H 2 ⁢ 3 1 + θ + α ⁡ α ⁡ 5 θ − 1 − 2 − 9 θ ⁢ w + x 1 + x 2 + 1 − α ⁢ 2 H 1 2 ⁢ x 2 + 4 H 1 ⁢ w + x 1 + x 2 ⁢ 2 H 1 − H 6 / 2 H 1 2 ,

where H 5 = 2 + θ + α θ α 3 + θ − 5 − 2 H 6 = 4 2 + θ + α θ α 11 + θ − 17 − 8 .

The following second -order conditions emerge:

∂ 2 O 1 M ∂ x 1 M 2 = H 2 + 1 − α α + 3 H 2 2 H 3 H 1 2 + 2 1 + α − θ H 5 H 1 − 3 + 4 α − 3 θ H 5 2 H 1 2 < 0 ∂ 2 O 2 M ∂ x 2 M 2 = ( α − 1 ( ⁢ 3 5 + 2 θ ⁢ 7 + 2 θ + α 4 ⁢ 1 + 3 θ ⁡ 4 + θ ⁢ 1 + θ ⁡ 14 + θ − 2 α ( 116 . + θ ⁡ 235 + 72 ⁢ θ ) 6 α 2 ⁢ 25 + θ ⁡ 117 + 86 ⁢ θ + 4 θ 2 − 2 α 3 ( 12 + θ ( 165 + θ ( 274 + 45 ⁢ θ ) ) ) ) ) / H 1 2 < 0

Thus, the second-order conditions are satisfied.

Moreover, the expressions represented by H_7-10 are as follows:

H 7 = α 5 ⁢ θ ⁡ 3 + θ ⁢ 1 + 3 θ ⁢ 1 + θ ⁡ 14 + θ − 2 + θ ⁢ 23 + θ ⁡ 23 + 5 θ + α ( 120 + θ ( 393 + 62 ⁢ θ ⁡ 5 + θ ) ) − 4 α 2 ⁢ 26 + θ ⁡ 159 + θ ⁡ 235 + θ ⁡ 82 + θ + 4 α 3 ( 8 + θ ( 98 + θ ( 263 + θ ⁡ 180 + 11 ⁢ θ ) ) ) + α 4 ( θ ( θ − 83 − 4 α 2 ( 26 + θ ⁡ 159 + θ ⁡ 235 + θ ⁡ 82 + θ − 2 ) H 8 = θ − 1 ⁢ θ ⁡ 1 + θ ⁡ 14 + θ H 9 = 2 56 + θ ⁡ 91 − θ ⁡ 47 + 25 ⁢ θ + 2 α ⁡ θ ⁡ θ ⁡ 2 θ ⁡ 40 + θ − 23 − 149 − 23 − 2 α 2 ( 4 + θ ⁡ θ ⁡ θ ⁡ 57 + 10 ⁢ θ − 99 − 44 ) − α 3 ( θ ⁡ θ ⁡ 54 + θ ⁡ 28 + θ − 17 ⁢ θ − 33 − 1 ) H 10 = α 2 ⁢ θ ⁡ 23 + 18 − 25 ⁢ θ ⁢ θ + α 3 ⁢ H 8 + 16 + θ ⁡ 33 + 8 θ + α ⁡ θ ⁡ θ ⁡ 8 θ − 9 − 47 − 8 .

Figure 1:

The parameter range that satisfies the stability conditions.

In Figure 1, the orange area represents the region that satisfies the stability conditions of the R&D reaction function system, determined by θ and α. Additionally, to ensure that the R&D cost condition is satisfied after the optimal R&D investment choice, we have (c–x_i-β_jx_j) > 0. The solution yields c > w 2 α 5 H 8 − 59 + θ θ 21 + 5 θ − 3 + α H 9 + ( ( 2 w H 2 2 2 + θ ) 2 − α H 10 ) ) / H 7 . All solutions satisfy the conditions mentioned above.

Section 4: Private Duopoly

First-order and second-order conditions for the equilibrium outputs:

The following first-order conditions emerge:

∂ O i P ∂ q i P = − q j − α − 1 w − 4 q i + x i + x j β j .

It is easy to verify that second-order conditions are satisfied.

First-order and second-order conditions for the equilibrium R&D levels:

The following first-order conditions emerge:

∂ O i P ∂ x i P = α − 1 2 α − 3 8 α − 7 x i + 2 α − 2 w + x j 5 − 4 α 2 .

It is easy to verify that second-order conditions are satisfied.

Furthermore, we need to ensure that the stability conditions of the R&D reaction function system, as well as the parameter range for the R&D cost conditions, are satisfied. To meet the stability conditions of the R&D reaction function system, we must ensure that ∂ x 1 P ∂ x 2 P < 1 and ∂ x 2 P ∂ x 1 P < 1 . The solution reveals that, regardless of the values of θ and α, the stability conditions of the R&D reaction function system are satisfied. Moreover, to ensure that the R&D cost condition is met after the optimal R&D investment is chosen, we have (c−x_i−β_jx_j) > 0. The solution yields c > 4w(2−α)/(17 + 4α(4α−9)). All solutions satisfy the above conditions.

Section 6 Extension 1: Unilateral Shareholding

The semi-public firm as the shareholder:

First-order and second-order conditions for the equilibrium outputs:

The following first-order conditions emerge:

∂ O 1 U 1 ∂ q 1 U 1 = w − 3 + θ q 1 − 1 + α q 2 + x 1 + x 2 β 2 ∂ O 2 U 1 ∂ q 2 U 1 = α − 1 q 1 − w + 4 q 2 − x 2 − x 1 β 1 .

It is easy to verify that second-order conditions are satisfied. Therefore, the following equilibrium outputs are obtained through solving:

q 1 U 1 = w α − 3 + x 1 1 + α β 1 − 4 + x 2 1 + α − 4 β 2 α − 4 θ − 11 q 2 U 1 = x 1 1 − 3 + θ β 1 + x 2 β 2 − 3 − θ − w 2 + θ α − 4 θ − 11 .

First-order and second-order conditions for the equilibrium R&D levels:

The following first-order conditions emerge:

∂ O 1 U 1 ∂ x 1 U 1 = − ( α − 1 ⁢ α 2 ⁢ 1 + θ − 59 − 6 α ⁡ 1 + θ ⁡ 3 + θ + θ ⁡ θ ⁡ 21 + 5 θ − 3 + ( 2 + θ ) ( 31 + α 2 − 2 α ⁡ 7 + 3 θ + θ ⁡ 27 + 5 θ ) x 1 + ( α 2 ⁢ 2 + θ − 28 − α ⁡ 5 + 2 θ ⁢ 4 + 3 θ + θ ( 1 3 + θ ⁡ 22 + 5 θ ) ) x 2 ) / α − 11 − 4 θ 2 ∂ O 2 U 1 ∂ x 2 U 1 = α − 1 ⁢ α − 6 θ − 17 ⁢ α − 2 θ − 5 ⁢ x 2 − 4 2 + θ ⁢ 3 + θ ⁢ w + x 1 / α − 11 − 4 θ 2 .

It is easy to verify that second-order conditions are satisfied. Therefore, the following equilibrium R&D levels are obtained through solving:

x 1 U 1 = w H 41 2 + θ H 42 , x 2 U 1 = − 4 w H 43 H 42

where H 41 = 517 + 201 θ + α 3 1 + θ − 5 θ 2 43 + θ 27 + 4 θ − α 2 17 + θ 33 + 10 θ + α 19 + θ 223 + θ 159 + 29 θ H 42 = 209 − α 3 + 5 α 2 5 + 2 θ − α 173 + 29 θ 5 + θ + θ 317 + θ 143 + 20 θ H 43 = α − 11 − 4 θ 3 + θ .

The private firm as the shareholder:

First-order and second-order conditions for the equilibrium outputs:

The following first-order conditions emerge:

∂ O 1 U 2 ∂ q 1 U 2 = 4 α − 1 θ − 3 q 1 − α θ − 1 w − q 2 + x 1 + x 2 β 2 ∂ O 2 U 2 ∂ q 2 U 2 = w − 1 + α θ q 1 − 4 q 2 + x 2 + x 1 β 1 .

It is easy to verify that second-order conditions are satisfied. Therefore, the following equilibrium outputs are obtained through solving:

q 1 U 2 = 1 − α θ ⁢ 3 w − x 1 ⁢ β 1 − 4 + x 2 ⁢ 4 β 2 − 1 / H 48 q 2 U 2 = ( w ⁡ 2 + θ + α θ ⁡ α θ − 4 + x 1 ⁢ α 2 ⁢ θ 2 − 1 + 3 + θ − 4 α θ ⁢ β 1 + x 2 ( 3 + θ − 4 α θ + α 2 ⁢ θ 2 − 1 ⁢ β 2 ) ) / H 48 ,

where

H 48 = 11 + θ 4 + α α θ − 16 .

First-order and second-order conditions for the equilibrium R&D levels:

The following first-order conditions emerge:

∂ O 1 U 2 ∂ x 1 U 2 = α θ − 1 ⁢ x 1 − 9 α θ − 1 2 ⁢ 4 α − 1 ⁢ θ − 3 ⁢ w + x 1 + x 2 / H 48 2 + 3 ( 2 + θ + α θ ( α θ − 4 ) ) H 49 / H 48 2 − 2 H 49 / H 48 ∂ O 2 U 2 ∂ x 2 U 2 = − 2 w H 50 + 2 H 50 ⁢ x 1 + 3 H 51 ⁢ x 2 / H 48 2 ,

where

H 49 = θ − 1 2 + θ + α θ α θ − 4 w + x 1 + x 2 H 50 = α θ 17 + θ 4 + α 5 + θ 8 + α α θ − 17 − 8 − 2 θ 4 + θ H 51 = 35 + 4 θ 6 + θ + α θ θ α 96 + θ 8 + α α θ − 22 − 40 − 106 .

It is easy to verify that second-order conditions are satisfied. Therefore, the following equilibrium R&D levels are obtained through solving:

x 1 U 2 = − w 59 + θ H 52 H 53 , x 2 U 2 = − 2 w α θ − 1 H 50 H 53

where H 52 = ( 3 − 194 α + α 46 + 193 α − 21 θ − 5 + α α 81 + 52 α − 40 θ 2 − α 2 ( 4 + ( α − 16 ) α ) θ 3 + α 4 θ 4 ) H 53 = θ ( 229 α − 69 + 7 34 − 59 α α − 33 θ + α 52 + α 302 α − 225 − 5 θ 2 − 2 α 2 ( 2 + 5 α 7 α − 4 ) θ 3 + α 4 1 + 3 α θ 4 ) − 46 .

Section 6 Extension 2: Mixed Duopoly with a Foreign-Owned Private Firm

First-order and second-order conditions for the equilibrium outputs:

The following first-order conditions emerge:

∂ O 1 F ∂ q 1 F = 1 + θ ⁡ 4 θ − 9 − 4 α ⁡ θ − 2 ⁢ q 1 + 1 − α ⁡ θ − 2 ⁢ θ − 1 + θ − 3 ⁢ θ ⁢ q 2 + α − 1 θ − 2 ⁢ θ ⁡ w + x 1 + x 2 ⁢ β 2 ∂ O 2 F ∂ q 2 F = α − 1 − α θ ⁢ q 1 − α − 1 ⁢ w − 4 q 2 + x 2 + x 1 ⁢ β 1 .

It is easy to verify that second-order conditions are satisfied. Therefore, the following equilibrium outputs are obtained through solving:

q 1 F = ( α − 1 ( ⁢ w ⁡ 1 + 5 − 3 θ ⁢ θ + α ⁡ θ − 2 ⁢ 1 + 3 θ + x 1 ( 4 θ − 1 ⁢ θ − 2 ⁢ θ + ( 1 − α ( θ − 2 ) θ − 1 + θ − 3 ⁢ θ ) β 1 ) + x 2 ( 1 − α ⁡ θ − 2 ⁢ θ − 1 + θ − 3 ⁢ θ + 4 α − 1 ( θ − 2 ) θ β 2 ) ) ) / H 50 q 2 F = ( α − 1 ( ⁢ w ⁡ θ ⁡ 7 − 3 θ + α ⁡ θ − 6 + θ 2 − 1 + x 1 ( 1 + α ⁡ θ − 1 ⁢ θ − 2 ⁢ θ + ( θ ( 9 − 8 α + 4 α − 1 ⁢ θ ) − 1 ) β 1 ) + x 2 ( θ ⁡ 9 − 8 α + 4 α − 1 ⁢ θ − 1 + 1 + α ⁡ θ − 1 ⁢ θ − 2 θ β 2 ) ) ) / H 50 .

First-order and second-order conditions for the equilibrium R&D disclosure levels:

The following first-order conditions emerge:

∂ O 1 F ∂ β 1 F = ( α − 1 ⁢ θ − 2 ⁢ x 1 ( w ( − α 4 ( θ − 2 ) 2 ⁢ θ ⁡ 3 + θ ⁢ 1 + θ ⁡ 14 + θ − θ ⁡ 5 + θ ⁡ 3 θ − 11 × ( 1 + θ ⁡ 4 θ − 9 ) + α 2 ⁢ θ ⁡ 102 + θ ⁡ θ ⁡ 111 + 7 31 − 10 ⁢ θ ⁢ θ − 586 − 2 + α 3 ⁢ θ − 2 ( 1 + θ ⁡ 14 + θ ⁡ θ ⁡ θ ⁡ 53 + θ − 5 − 272 ) + α ( 4 + θ ( θ ⁡ 130 + θ ⁡ 174 + θ ⁡ 44 ⁢ θ − 199 − 56 ) ) ) − x 1 ⁢ θ H 64 + H 65 ⁢ H 66 ⁢ β 1 − x 2 ⁢ H 65 ⁢ H 66 + θ H 64 ⁢ β 2 ) ) / H 67 2 ∂ O 2 F ∂ β 2 F = ( α − 1 ⁢ θ x 2 ( w ( α 4 ⁢ θ − 2 2 ⁢ 1 + 3 θ ⁢ 1 + θ ⁡ 14 + θ + 4 θ − 2 ⁢ 1 + θ ⁡ 3 θ − 7 − α 3 θ − 2 ⁢ 4 + θ ⁡ θ ⁡ 3 θ ⁡ 41 + 2 θ − 236 − 105 + α ( 27 − θ ( 198 + θ ( 43 + θ ( 49 ⁢ θ − 166 ) ) ) ) + α 2 ⁢ θ ⁡ 287 + θ ⁡ 357 + θ ⁡ 3 θ ⁡ 43 + θ − 517 − 31 ) + x 1 ( 4 α − 1 ( θ − 2 ) θ H 68 + H 69 + H 70 + H 71 + H 72 ⁢ β 1 ) + x 2 ( H 69 + H 70 + H 71 + H 72 + 4 α − 1 ( θ − 2 ) θ H 68 ⁢ β 2 ) ) ) / H 67 2 ,

where H 64 = ( α 4 θ − 2 2 θ − 1 1 + θ 14 + θ + 3 + θ 4 θ − 11 1 + θ 4 θ − 9 − α 3 θ − 2 × 1 + θ 101 + θ θ 53 + θ − 172 + α θ 153 + 2 θ 10 θ − 51 − 12 + α 2 ( 11 + θ 2 θ 342 + θ 43 θ − 221 − 291 ) ) H 65 = 9 + 4 α θ − 2 − 4 θ θ − 1 H 66 = θ 3 − 2 θ + α 3 θ − 2 1 + θ 14 + θ + α 2 θ 63 + θ − 30 θ − 2 + α ( 4 + θ ( 3 ( 6 − θ ) θ − 32 ) ) H 67 = 3 + 3 θ 5 θ − 11 + α 2 θ − 2 1 + θ 14 + θ − α 1 + θ θ 27 + θ − 61 H 68 = ( θ − 2 + α ( 12 + θ 16 θ − 41 + α 2 θ − 2 1 + θ θ − 2 − α ( 8 + θ ( θ 28 + θ − 69 ) ) ) ) H 69 = 4 θ − 2 1 + θ 4 θ − 9 − α 4 θ − 2 2 θ − 1 1 + θ 14 + θ H 70 = α 3 θ − 2 θ 129 + θ θ 37 + 2 θ − 148 − 4 H 71 = α 27 + θ θ 349 + 5 θ 3 θ − 26 − 310 H 72 = − α 2 31 + θ θ 603 + θ θ 39 + θ − 275 − 447 .

It is easy to verify that second-order conditions are satisfied. Therefore, the equilibrium R&D disclosure levels are β 1 F = β 2 F = 1 .

First-order and second-order conditions for the equilibrium R&D levels:

The following first-order conditions emerge:

∂ O 1 F ∂ x 1 F = ( α − 1 ⁢ 2 − θ ⁢ θ H 67 2 ⁢ x 1 + 2 ( α − 1 ) 2 ⁢ − H 65 ⁢ H 75 2 ⁢ w + x 1 + x 2 + α − 1 2 ⁢ H 75 2 ( − H 65 ) w + x 1 + x 2 + α − 1 2 ⁢ 1 + 4 α ⁡ θ − 2 − θ ⁢ H 74 2 ⁢ w + x 1 + x 2 − 2 1 − α ⁢ α 2 − θ ⁢ H 74 ⁢ H 73 ⁢ w + x 1 + x 2 ) / H 73 2 ∂ O 2 F ∂ x 2 F = ( ( 4 α − 1 ⁢ α θ H 77 ⁢ 1 + 5 − 3 θ ⁢ θ + α ⁡ θ − 2 ⁢ 1 + 3 θ ⁢ w + x 1 + x 2 − 1 − α × ( 2 H 73 2 ⁢ x 2 + 8 α − 1 ) 2 ⁢ H 74 2 ⁢ w + x 1 + x 2 − 4 1 − α ⁢ H 74 ⁢ H 76 ⁢ w + x 1 + x 2 ) / 2 H 73 2 ,

where

H 73 = 3 + 3 θ ⁡ 5 θ − 11 + α 2 ⁢ θ − 2 ⁢ 1 + θ ⁡ 14 + θ − α ⁡ 1 + θ ⁡ θ ⁡ 27 + θ − 61 H 74 = θ ⁡ 7 − 3 θ + α ⁡ θ − 6 + θ 2 − 1 H 75 = 1 + 5 − 3 θ ⁢ θ + α ⁡ θ − 2 ⁢ 1 + 3 θ H 76 = 4 + α 2 ⁢ θ − 2 ⁢ θ ⁡ 11 + θ + 4 θ ⁡ 3 θ − 7 − α ⁡ 4 + θ ⁡ θ ⁡ 21 + θ − 51 H 77 = 4 + 2 α − 3 ⁢ α − 16 ⁢ θ + 28 − 11 ⁢ α ⁢ α θ + α − 1 ⁢ 5 α − 6 ⁢ θ 2 .

It is easy to verify that second-order conditions are satisfied. Therefore, the following equilibrium R&D levels are obtained through solving:

x 1 F = ( w ( 2 α 4 ⁢ θ − 2 2 ⁢ θ − 1 ⁢ θ ⁡ 1 + θ ⁡ 14 + θ + 4 θ ⁡ θ − 2 ⁢ θ ⁡ 3 θ ⁡ 3 θ − 4 − 8 − 2 − α 3 θ − 2 ⁢ θ ⁡ θ ⁡ 162 + θ ⁡ 3 θ ⁡ 41 + θ − 304 − 15 − 1 + α ( 4 − 2 θ ( 22 + θ ( θ ( 253 + θ ⁡ 75 ⁢ θ − 277 ) − 14 ) ) ) + α 2 ( θ ⁡ 91 + θ ⁡ θ ⁡ 1010 + θ ⁡ θ ⁡ 213 + θ − 864 − 337 − 2 ) ) ) / H 78 x 2 F = ( 2 w θ ( − α 4 ( θ − 2 ) 2 ⁢ θ − 1 ⁢ θ ⁡ 1 + θ ⁡ 14 + θ + 2 ( ( 1 + θ ⁡ 3 θ − 7 ⁢ ) 2 + α 3 ⁢ θ − 2 ⁢ θ × ( θ ( 1 29 + θ ⁡ θ ⁡ 37 + 2 θ − 148 ) − 4 ) + α ( θ ⁡ 75 + θ ⁡ θ ⁡ 385 + θ ⁡ 15 ⁢ θ − 134 − 394 − 4 ) − α 2 ⁢ θ ⁡ 59 + θ ⁡ θ ⁡ 623 + θ ⁡ θ ⁡ 39 + θ − 277 − 499 − 2 ) ) / H 78 ,

where

H 78 = α 4 ( θ − 2 ) 2 ⁢ θ ⁡ 3 + θ ⁢ 1 + 3 θ ⁢ 1 + θ ⁡ 14 + θ + θ ⁡ 1 + θ ⁡ 3 θ − 7 ( 13 + θ ( 51 ⁢ θ − 115 ) ) − α 3 ⁢ θ − 2 ⁢ 1 + θ ⁡ 17 + θ ⁡ θ ⁡ θ ⁡ 217 + 3 θ ⁡ 51 + 2 θ − 1090 − 264 + α ( 2 θ ( 2 3 + θ ⁡ 127 − θ ⁡ 1462 + θ ⁡ θ ⁡ 163 + 30 ⁢ θ − 1141 ) − 4 ) + α 2 ( 2 + θ ( θ ( 303 + θ ( 35 61 + θ ⁡ θ ⁡ 134 + θ ⁡ 161 + 3 θ − 2778 ) ) − 106 ) ) ) .

Appendix B

This appendix presents the proofs of the main results in the main text (Tables 2–5).

Proof of Proposition 1.

We substitute Eqs. (5) and (6) into Eqs. (3) and (4) and obtain the following. First,

∂ O 1 M ∂ β 1 M = ( ( x 1 ⁢ α H 3 ⁢ 3 4 + θ + α ⁡ α − 13 + α θ ⁡ 14 + θ − θ ⁡ 18 + θ ⁢ w + x 2 + x 1 ⁢ β 1 − ( α − 1 ) α + θ − α θ ⁢ H 3 ⁢ w ⁡ α − 4 + θ + 3 α θ + x 1 ⁢ 4 H 2 + α + θ − α θ ⁢ β 1 + x 2 ( ( α + θ − α θ ) + 4 H 2 ⁢ β 2 ) − 4 α − 1 + θ ⁢ H 3 ( w H 5 + x 1 ⁢ α − 1 + α 2 ⁢ θ − 1 ⁢ θ + H 3 ⁢ β 1 + x 2 ( H 3 + α − 1 + α 2 ⁢ θ − 1 ⁢ θ ⁢ β 2 ) ) + α ( 3 4 + θ + α ⁡ α − 13 + α θ ⁡ 14 + θ − θ ⁡ 18 + θ ( w H 5 + x 1 ⁢ α − 1 + α 2 ⁢ θ − 1 ⁢ θ + H 3 ⁢ β 1 + x 2 ⁢ H 3 + α − 1 + α 2 ⁢ θ − 1 ⁢ θ ⁢ β 2 ) ) ) ) / H 11 2 ∂ O 2 M ∂ β 2 M = ( 1 − α ⁢ x 2 ( w ( α ( 16 + θ ⁡ 56 + 28 − 3 θ ⁢ θ − α 3 ⁢ θ ⁡ 1 + 3 θ ⁢ 1 + θ ⁡ 14 + θ + α 2 ⁢ θ ( 25 + 3 θ ⁡ 36 + θ ⁡ 23 + 2 θ ) + α ⁡ θ ⁡ θ ⁡ θ − 23 ⁢ θ − 122 − 76 − 8 ) − 4 2 + θ ) + x 1 ( − 4 H 2 H 12 + ( − 4 3 + θ + α ( 24 + θ ⁡ 24 + 4 − 3 θ ⁢ θ + α 3 ⁢ θ − 1 ⁢ θ ⁡ 1 + θ ⁡ 14 + θ + α 2 ⁢ θ ( 25 + θ ⁡ 12 − θ ⁡ 19 + 2 θ ) + α ⁡ θ ⁡ θ ⁡ θ ⁡ 9 + θ − 2 − 44 − 12 ) ) β 1 ) + x 2 ( − 4 3 + θ + α 4 + α ⁡ 12 − 13 ⁢ α ⁢ α − 2 ⁢ θ 2 − 12 ⁢ α − α ⁡ 44 + α − 25 ⁢ α ⁢ θ + α − 1 ( 3 + α ( − 6 + 13 α ) ) θ 3 + α − 1 ) 2 ⁢ α θ 4 + 24 ⁢ 1 + θ − 4 H 2 ⁢ H 12 ⁢ β 2 ) / H 11 2 ,

where

H 11 = 3 4 + θ + α θ 14 + θ − θ 18 + θ ) H 12 = 1 + α α − 2 + α − 9 α − 1 θ + 6 + 7 α 2 α − 3 θ 2 + α − 2 α θ 3 .

Therefore, the equilibrium disclosure of R&D knowledge is β 1 M = − w + x 2 / x 1 < 0 and β 2 M = − w + x 1 / x 2 < 0 . Additionally, the second-order conditions are ∂ 2 O 1 M / ∂ β 1 M 2 = x 1 1 − α H 3 α + θ − α θ 2 x 1 − H 3 2 4 α − 1 + θ x 1 + 2 α H 3 H 11 x 1 / H 11 2 > 0 ; ∂ 2 O 2 M / ∂ β 2 M 2 = 4 α − 1 H 2 H 12 x 2 2 / H 11 2 > 0 . Accordingly, both semi-public and private firms fully disclose their information with cross-ownership in a mixed duopoly β 1 M = β 2 M = 1 .

Proof of Lemma 1.

(i) First, because ∂ x 1 M / ∂ x 2 M = ( 2 α 5 θ − 1 θ 1 + θ 14 + θ − 59 + θ θ 21 + 5 θ − 3 + 2 α 56 + θ 91 − θ 47 + 25 θ + 2 α 2 θ θ 2 θ 40 + θ − 23 − 149 − 23 − 2 α 3 4 + θ θ θ 57 + 10 θ − 99 − 44 − α 4 θ θ 54 + θ 28 + θ − 17 θ − 33 − 1 ) / ( α 5 θ 3 + θ 12 + θ 1 + θ 14 + θ − 2 + θ 31 + θ 27 + 5 θ ) + α ( 152 + θ 475 + 6 θ 57 + 11 θ − 2 α 2 60 + θ 381 + 2 θ 256 + θ 82 + θ + 2 α 3 16 + θ 227 + θ 591 + 2 θ 172 + 7 θ + α 4 θ θ θ θ − 58 θ − 584 − 520 − 85 − 2 ) > 0 , ∂ x 2 M / ∂ x 1 M = − ( ( 2 − 2 2 + θ ) 2 + α 3 θ 23 + 18 − 25 θ θ + + α 4 θ − 1 θ 1 + θ 14 + θ + α 16 + θ 33 + 8 θ + α 2 θ θ 8 θ − 9 − 47 − 8 / ( 3 5 + 2 θ 7 + 2 θ + α 4 1 + 3 θ 4 + θ 1 + θ 14 + θ − 2 α 116 + θ 235 + 72 θ + 66 α 2 25 + θ 117 + 86 θ + 4 θ 2 − 2 α 3 ( 12 + θ 165 + θ 274 + 45 θ ) ) ) > 0 , x 1 M increases with x 2 M , while x 2 M increases with x 1 M . Thus, the R&D levels of firms are strategic substitutes for a mixed duopoly.

According to Eqs. (7) and (8), we obtain x 1 M − x 2 M = w θ − 1 H 13 / H 7 > 0 , where H₁₃=43+5θ(6 + θ)+2α⁵θ(1+θ)(1+θ(14+θ))−2α(40+9θ(10+3θ))+2α²(15+θ(101+2θ(41+θ)))−2α³(−4+θ(9+θ(43+2θ)))−α⁴(1+θ(36+θ(54+θ(36+θ)))). Therefore, x 1 M > x 2 M , and the semi-public firm invests more in R&D than dose the private firm.

According to Eqs. (7) and (8) and w>0, 0<α<0.5 0<θ<0.5, we obtain ∂ x 1 M / ∂ α = w 2 H 7 H 14 − H 15 H 16 / H 7 2 > 0 and ∂ x 2 M / ∂ α = 2 w H 17 / H 7 2 > 0 , where H 14 = 56 + 5 α 4 θ − 1 θ 1 + θ 14 + θ + θ 91 − θ 47 + 25 θ + 2 α ( θ ( θ 2 θ 40 + θ − 23 − 2 3 ) − 149 ) ) − 3 α 2 4 + θ θ θ 57 + 10 θ − 99 − 44 − 2 α 3 ( θ ( θ ( 54 + θ 28 + θ − 17 θ ) − 33 ) − 1 ) ) H 15 = 120 + θ 393 + 62 θ 5 + θ + 5 α 4 θ 3 + θ 1 + 3 θ 1 + θ 14 + θ − 8 α ( 26 + θ ( 159 + θ 235 + θ 82 + θ ) ) + 12 α 2 8 + θ 98 + θ 263 + θ 180 + 11 θ + 4 α 3 ( θ ( θ ( θ ( ( θ − 110 ) θ − 574 ) − 448 ) − 83 ) − 2 ) H 16 = 2 α 5 θ − 1 θ 1 + θ 14 + θ − 59 + θ θ 21 + 5 θ − 3 + 2 α 56 + θ 91 − θ 47 + 25 θ + 2 α 2 θ θ 2 θ 40 + θ − 23 − 149 − 23 − 2 α 3 4 + θ θ θ 57 + 10 θ − 99 − 44 − α 4 ( θ θ 54 + θ 28 + θ − 17 θ − 33 − 1 )

H 17 = − 2 α ⁡ 2 + θ ⁢ 232 + θ ⁡ 1119 + θ ⁡ 2577 + θ ⁡ 1911 + 462 ⁢ θ + 8 θ 2 − 2 + θ ( θ ( θ ( θ ( 2 θ ( 1 + 5 θ ) − 249 ) − 635 ) − 501 ) − 112 ) − 2 α 7 ⁢ θ 2 ⁢ 1 + θ ⁡ 14 + θ ( 61 + θ ( 145 + θ ( 35 + θ ( 343 + 4 θ ⁡ 137 + 5 θ ) ) ) ) − α 8 ⁢ θ 2 ⁢ 1 + θ ⁡ 14 + θ ( − 1 + θ ( θ ⁡ θ ⁡ 46 + θ ⁡ 219 ⁢ 33 − θ ⁢ θ − 13 − 37 ) ) + α 2 ( 1472 + θ ( 9634 + θ ( 29962 + θ ( 48204 + θ ( 35062 + θ ( 12329 + 32 θ ( 63 + 4 θ ) ) ) ) ) ) ) − 2 α 3 ( 544 + θ ( 5132 + θ ( 18487 + θ ( 38943 + 2 θ ( 22534 + θ ( 13220 + θ ( 3853 + 420 ⁢ θ ) ) ) ) ) ) ) + α 6 ⁢ θ ( 102 + θ ( 2662 + θ ( 11673 + θ ( 17676 + θ ( 27624 + θ ( 49402 + θ ( 279 17 + 20 ⁢ θ ⁡ 71 + θ ) ) ) ) ) ) ) + α 4 ( 352 + θ ( 5918 + θ ( 27620 + θ ( 65777 + θ ( 104099 + θ ( 10 0931 + θ ⁡ 49012 + θ ⁡ 8656 + 51 ⁢ θ ) ) ) ) ) ) − 2 α 5 ( 16 + θ ( 782 + θ ( 6187 + θ ( 17072 + θ ( 2 8808 + θ ⁡ 42568 + θ ⁡ 36941 + 2 θ ⁡ 5501 + 120 ⁢ θ ) ) ) ) ) .

Thus, both semi-public and private firms’ R&D levels increase with cross-ownership.

We substitute β 1 M = β 2 M = 1 and Eqs. (7) and (8) into Eqs. (5) and (6), respectively, and obtain ∂ q 1 M / ∂ α = w H 18 / H 7 2 > 0 and ∂ q 2 M / ∂ α = w H 19 / H 7 2 > 0 , where

H 18 = 40 ⁢ 7 − α ( α − 1 ) 4 + ( α − 1 ) 2 ( 2357 + α ( α ( 6343 + α ( α ⁡ 419 + α − 30 ⁢ α − 286 8 ) ) − 5902 ) ) θ + 2 α − 1 ( − 702 + α ( 8465 + α ( α ( 22254 + α ( α ( 3766 + α ( 17 ⁢ α − 410 ) ) − 13149 ) ) − 20101 ) ) ) θ 2 + ( − 976 + α ( 1058 + α ( 30456 + α ( α ( 124752 + α α ⁡ 35799 + α ⁡ 391 ⁢ α − 6702 − 89764 ) − 94986 ) ) ) ) θ 3 + 2 ( α ( 4207 + α ( α ( 897 0 + α ⁡ 13914 + α ⁡ α ⁡ 19433 + 412 ⁢ α ⁡ 2 α − 17 − 26212 − 13837 ) ) − 379 ) ) θ 4 + α ( 2536 + α ( α ( 51856 + α ⁡ α ⁡ 48028 + α ⁡ α ⁡ 2884 + 1083 ⁢ α − 21686 − 6683 9 ) ) − 17341 ) − 120 ) θ 5 − 2 α ( α ( 909 + α ( α ( 13506 + α ( α ( 12712 + 39 α ( 45 ⁢ α − 18 7 ) ) − 16195 ) ) − 5092 ) ) − 60 ) θ 6 + α 3 ( 60 + α ( α ( 1024 + α ( 1383 + α ( 145 ⁢ α − 181 4 ) ) ) − 489 ) ) θ 7 + 2 α 5 ⁢ 24 + α ⁡ 2 α ⁡ 49 ⁢ α − 51 − 69 ⁢ θ 8 + 12 ⁢ α 8 ⁢ θ 9 H 19 = α 8 ⁢ θ − 1 ⁢ θ 2 ⁢ 3 + θ ⁢ 1 + θ ⁡ 14 + θ ⁢ θ ⁡ θ ⁡ θ − 31 ⁢ θ − 96 − 47 − 3 − ( 2 + θ ( θ ( θ θ ⁡ 212 + θ ⁡ 139 + 20 ⁢ θ − 244 − 594 ) − 262 ) + 2 α 7 ⁢ θ ⁡ 3 + θ ⁢ 1 + θ ⁡ 14 + θ ( θ ( θ ⁡ θ ⁡ θ ⁡ 223 + 29 ⁢ θ − 30 − 199 − 69 ) − 2 ) + 2 α ⁡ 2 + θ ( θ ( θ ( θ ( 549 + θ ( 949 + 2 09 ⁢ θ ) ) − 2128 ) − 2030 ) − 546 ) + 2 α 3 ( θ ( θ ( θ ( θ ( 3318 + θ ( 25667 + 8726 θ + 402 θ 2 ) ) − 43334 ) − 41197 ) − 13498 ) − 1320 ) + α 2 ( 3500 + θ ( 22498 − θ ( θ ( θ ( 1598 6 + θ ⁡ 16751 + θ ⁡ 3669 + 59 ⁢ θ ) − 27273 ) − 46293 ) ) ) + α 4 ( 900 + θ ( 16502 + θ ( 78283 − θ ⁡ θ ⁡ θ ⁡ 72961 + θ ⁡ 47219 + 4539 ⁢ θ + 57 ⁢ θ 2 − 40627 − 130448 ) ) ) + 2 α 5 ( θ ( θ ( θ ⁡ θ ⁡ θ ⁡ 24288 + θ ⁡ 32666 + θ ⁡ 6434 + 249 ⁢ θ − 38969 − 51960 − 194 78 ) − 2330 ) − 52 ) + α 6 ( 4 + θ ( 430 − θ ( θ ( θ ( θ ( 6451 + θ ( 49249 + θ ( 15958 + θ ( 1 541 + 17 ⁢ θ ) ) ) ) − 59473 ) − 42100 ) − 8841 ) ) )

Therefore, both semi-public and private firms’ outputs increase with cross-ownership.

Proof of Lemma 2.

(i) We substitute Eqs. (5) and (6), β 1 M = β 2 M = 1 and Eqs. (7) and (8) into Eq. (1), and obtain ∂ π 1 M / ∂ α = w 2 H 20 / H 7 3 < 0 and ∂ π 2 M / ∂ α = w 2 H 21 / H 7 3 > 0 , where

H 20 = α − 1 ) 4 ( 223409 + α ( α ( 996840 + α ( α ⁡ α ⁡ 59830 + α ⁡ α ⁡ 82 + α − 4240 − 283706 − 18434 ) ) − 949302 ) ) θ − 4 ( α − 1 ) 6 ⁢ α ⁡ α ⁡ 1794 + α ⁡ 7 α − 194 − 4194 − 15469 − 2 ( α − 1 ) 3 ( α ( α ( 2836560 + α ( α ( 1069157 + α ( 967992 + α ( α ⁡ 92566 − 4773 ⁢ α + 69 ⁢ α 2 − 638 067 ) ) − 3614605 ) − 37029 ) − 695182 ) ) ) θ 2 + ( α − 1 ) 2 ( α ( 3555619 + α ( α ( α ( 15867795 + α ( α ⁡ α ⁡ 8707531 + α ⁡ α ⁡ 284246 + α − 10063 + 92 ⁢ α − 2805109 − 7803657 − 718 6307 ) ) − 5308297 ) − 4811756 ) ) ) − 464582 ) θ 3 + α − 1 ( 276109 + α ( α ( 30601591 + α ( 2 α ( 48189769 + α ( α ( 20956126 + α ( α ( 17301807 + α ( α ( 1567845 + α ( 1937 ⁢ α − 10 4127 ) ) − 7795604 ) ) − 20374668 ) ) − 34505780 ) ) − 76175821 ) ) − 5172113 ) ) θ 4 + ( 300 96 + α ( 1058460 + α ( − 18588813 + α ( 108096133 + α ( − 306299205 + α ( 498851442 + α ( α ( 357472056 + α ( α ( 104737790 + α ( α ⁡ 12894315 + α ⁡ 56728 ⁢ α − 1663027 − 463 24317 ) ) − 199825770 ) ) − 510495768 ) ) ) ) ) ) ) θ 5 + ( 80836 + α ( − 1110278 + α ( 3765573 + α ( 13623346 + α ( α ( 400695918 + α ( 2 α ( 314098679 + α ( α ( 88700659 + α ( α ( 10345 134 + α − 2109046 + 159849 ⁢ α ) − 32616820 ) ) − 198773394 ) ) − 643210685 ) ) − 13345 5224 ) ) ) ) ) ) θ 6 + ( 30587 + α ( − 672490 + α ( 5811207 + α ( α ( 42726418 + α ( 19834908 + α ( α ( 392751658 + 3 α ( α ( 68973626 + α ( α ⁡ 5304396 + α ⁡ 149592 ⁢ α − 1106245 − 230 86692 ) ) − 124450542 ) ) − 213635889 ) ) − 24199356 ) ) ) ) ) θ 7 − ( − 4920 + α ( 142324 + α α ( 11295868 + α ( 2 α ( 46001481 + α ( α ( 4380276 + α ( 46504212 + α ( 2 α ( 12837800 + α ⁡ α ⁡ 76106 + 16767 ⁢ α − 2655918 ) − 52569845 ) ) ) − 48347165 ) ) − 42748687 ) ) − 172 6518 ) ) ) θ 8 + ( 300 + α ( − 10920 + α ( 177930 + α ( − 1624948 + α ( 8987576 + α ( − 30811 938 + α ( 64243522 + α ( − 75755440 + α ( 41252721 + α ( 4381246 − α ( 18555376 + 3 α ( α ⁡ 428399 + 25600 ⁢ α − 3022000 ) ) ) ) ) ) ) ) ) ) ) ) θ 9 + α 2 ( 1800 + α ( − 42276 + α ( 433626 + α − 2478570 + α ( 8426205 + 2 α ( α ( 9050910 + α ( α ( α ( 2433636 + α ( 135034 ⁢ α − 100146 1 ) ) − 1495660 ) − 3870476 ) ) ) − 8415635 ) ) ) ) ) θ 10 + α 4 ( 2223 + α ( α ( 281019 + α ( α ( 2190 750 + α ⁡ α ⁡ 872180 + α ⁡ 257700 + α ⁡ 86204 ⁢ α − 337805 − 2254057 ) ) − 1068856 ) − 38718 ) ) θ 10 + α 4 ( 2223 + α ( α ( 281019 + α ( α ( 2190750 + α ( α ( 872180 + α ( 257700 + α 86204 ⁢ α − 337805 ) ) − 2254057 ) ) ) − 1068856 ) ) − 38718 ) θ 11 + 2 α 6 ( 408 + α ( α ( 25497 + α ⁡ α ⁡ 47599 + α ⁡ α ⁡ 3469 ⁢ α − 7636 − 8736 − 53408 ) − 5316 ) ) θ 12 + α 8 ( 18 + α ( 150 + α ⁡ α ⁡ 1503 + α ⁡ 72 ⁢ α − 1313 − 1209 ) ) θ 13 − 6 α 10 ⁢ 12 + α ⁡ α ⁡ 8 + α − 30 ⁢ θ 14 + 3 α 12 ⁢ θ 15

H 21 = 2 + θ 2 ⁢ − 9736 + θ ⁡ − 29765 + θ ⁡ − 20493 + θ ⁡ 5497 + 4 θ ⁡ 2261 + 12 ⁢ θ ⁡ 56 + 5 θ − 2 α 13 ⁢ θ 4 ( 1 + θ ⁡ 14 + θ ⁢ ) 2 ( − 46 + θ ( − 649 + θ ( − 1084 + θ ( 293 + θ ( 258 + θ ( − 703 + 3 ( − 40 + θ ) θ ) ) ) ) ) ) − α ⁡ 2 + θ ( − 122288 + θ ( − 630080 + θ ( − 1111621 + 2 θ ( − 296958 + θ ( 875 86 + θ ⁡ 139301 + 2 θ ⁡ 23903 + 12 ⁢ θ ⁡ 243 + 5 θ ) ) ) ) ) + 3 α 2 ⁢ 2 + θ ( − 108448 + θ ( − 7865 06 + θ ( − 2015968 + θ ( − 2054040 + θ ( − 350819 + θ ( 594215 + θ ( 352229 + 8 θ ( 8377 + 426 ⁢ θ ) ) ) ) ) ) ) ) − 2 α 12 ⁢ θ 3 ( 1 + θ ( 14 + θ ( 147 + θ ( 5856 + θ ( 63572 + θ ( 171211 + θ ( 106912 + θ ⁡ − 48251 + θ ⁡ 39704 + θ ⁡ 70245 + θ ⁡ 8321 + 75 ⁢ θ ) ) ) ) ) + 3 α 11 ⁢ θ 2 ( 1 + θ ( 14 + θ ( 114 + θ ( 6920 + θ ( 107674 + θ ( 444917 + θ ( 624572 + θ ( 162415 + θ ( − 49494 + θ ( 202003 + θ ⁡ 119126 + 3 θ ⁡ 1931 + 8 θ ) ) ) ) ) ) ) ) + α 3 ( 948928 + θ ( 10101840 + θ ( 39028438 − θ ( − 70048150 + θ ( − 56137596 + θ ( − 8349947 + 2 θ ( 7176697 + 3 θ ( 1475953 + 2 θ ( 161191 + 6 θ ⁡ 2047 + 24 ⁢ θ ) ) ) ) ) ) ) ) ) + α 4 ( − 818432 + θ ( − 12185960 + θ ( − 63569092 + θ ( − 1544 70260 + θ ( − 182660329 + θ ( − 84579259 + θ ( 16227862 + θ ( 28674748 + θ ( 9144617 + θ ⁡ 984695 + 20436 ⁢ θ ) ) ) ) ) ) ) ) ) + 6 α 5 ( 69824 + θ ( 1513816 + θ ( 10900006 + θ ( 356893 28 + θ ( 58196933 + θ ( 44358248 + θ ( 8116872 + θ ( − 7674931 + θ ( − 4028180 + θ ( 606 θ 2 − 545611 − 4515 ⁢ θ ) ) ) ) ) ) ) ) ) ) − α 6 ( 120896 + θ ( 4100408 + θ ( 42615256 + θ ( 1919284 76 + θ ( 425038902 + θ ( 464402746 + θ ( 205049175 + θ ( − 15684005 + θ ( − 33422380 + θ ⁡ − 4261464 + θ ⁡ 706135 + 66879 ⁢ θ ) ) ) ) ) ) ) ) ) + 2 α 7 ( 8864 + θ ( 523800 + θ ( 8492850 + θ ( 55098379 + θ ( 168575540 + θ ( 254746251 + θ ( 175356573 + θ ( 33166544 + θ ( − 8 445718 ⁢ θ ⁡ 2256331 + 3 θ ⁡ 825069 + θ ⁡ 86773 + 364 ⁢ θ ) ) ) ) ) ) ) ) ) − 3 α 8 ( 416 + θ ( 44620 + θ ( 1270996 + θ ( 12900451 + θ ( 57127260 + θ ( 120810006 + θ ( 118933684 + θ ( 436165 88 + θ ⁡ 3222792 + θ ⁡ 7061566 + 3 θ ⁡ 1689776 + 9 θ ⁡ 27139 + 396 ⁢ θ ) ) ) ) ) ) ) ) − α 10 ⁢ θ ( 17 2 + θ ( 20346 + θ ( 679132 + θ ( 8712614 + θ ( 43510909 + θ ( 92719300 + θ ( 77512612 + θ ⁡ 13788780 + θ ⁡ 9042218 + θ ⁡ 21171490 + θ ⁡ 7454584 + θ ⁡ 459790 + 7221 ⁢ θ ) ) ) ) ) ) ) + α 9 ( 32 + θ ( 8000 + θ ( 415938 + θ ( 7535531 + θ ( 52710874 + θ ( 163507618 + θ ( 231 029362 + θ ( 127989330 + θ ( 17925294 + θ ( 23870476 + θ ( 24681884 + 3 θ ( 1788209 + 6 θ ⁡ 9700 + 57 ⁢ θ ) ) ) ) ) ) ) ) ) ) ) )

Therefore, cross-ownership decreases the profits of the semi-public firm while increasing the profits of the private firm.

We substitute Eqs. (5) and (6), β 1 M = β 2 M = 1 and Eqs. (7) and (8) into Eq. (2), respectively, and obtain ∂ P S M / ∂ α = w 2 H 22 / H 7 3 < 0 , ∂ C S M / ∂ α = w 2 H 2 H 23 H 1 H 24 / H 7 3 > 0 and ∂ S W M / ∂ α = w 2 H 25 / H 7 3 > 0 , where

H 22 = 4 ( α − 1 ) 6 ⁢ 5733 − α ⁡ α ⁡ 2058 + α ⁡ 7 α − 202 − 6922 − + ( α − 1 ) 4 ( 65405 + α ( α ( α ( 570286 + α ⁡ α ⁡ 67142 + α ⁡ α ⁡ 82 + α − 4412 − 387286 ) − 97788 ) ) − 198870 ) ) θ − 2 ( α − 1 ) 3 ( α ( 415327 + α ( α ( 1722751 + α ( α ( 2309907 + α ( α ( 102226 − 4944 α + 69 ⁢ α 2 ) − 816543 ) ) − 2852860 ) ) − 743424 ) ) − 142413 + ) ) θ 2 + ( α − 1 ) 2 ( α ( 5675574 + α ( α ( 34534928 + α ( α ( 45623786 + α ( α ( 14960266 + α ( α ( 309206 + α ( 92 ⁢ α − 103 57 ) ) − 3434027 ) ) − 33370622 ) ) − 44873530 ) ) − 18854241 ) ) − 554331 ) ) θ 3 + 2 ( α − 1 ) ( 119219 + α ( − 2726678 + α ( 19293691 + α ( − 62163813 + α ( 115444031 + α ( α ( 126139040 + α ( α ⁡ 39600071 + α ⁡ α ⁡ 1866979 + α ⁡ 1983 ⁢ α − 111995 − 11852777 − 83767294 ) ) − 141842317 ) ) ) ) ) ) θ 4 + ( 82521 + α ( 326084 + α ( α ( 116446080 + α ( α 765000930 + α ( α ( 866964558 + α ( α ( 268245408 + α ( α ( 18772134 + α ( 60602 ⁢ α − 1954433 ) ) − 89835226 ) ) − 562255788 ) ) − 974898514 ) ) − 390878464 ) ) − 16075 980 ) ) ) ) θ 5 + 2 ( 50796 + α ( α ( 3830796 + α ( α ( α ( 224698575 + α ( α ( 489455252 + α α ⁡ 204215340 + α ⁡ α ⁡ 20786760 + α ⁡ 188213 ⁢ α − 3176121 − 78976470 − 37717 3920 ) ) − 424129930 ) ) − 58613681 ) ) − 365024 ) ) − 790052 ) ) θ 6 + ( 34235 + α ( − 791 430 + α ( 7269990 + α ( α ( 71401166 + α ( α ( α ( 459084746 + α ( α ( 334910208 + α ( α ⁡ 43967208 + α ⁡ 768474 ⁢ α − 8453611 − 146772688 ) ) − 504201390 ) ) − 1979518 84 ) ) − 26214678 ) ) − 33055074 ) ) ) θ 7 + 2 ( 2580 + α ( α ( 974007 + α ( α ( 25946652 + α ( α ( 65058355 + 2 α ( α ( α ( 30766246 + α ( α ⁡ 4792584 + α ⁡ 95427 ⁢ α − 885970 − 1 6284995 ) ) − 25669200 ) − 6412997 ) ) − 58086021 ) ) − 6615080 ) ) − 77234 ) ) θ 8 + ( 3 00 + α ( − 11160 + α ( 188154 + α ( α ( 9972271 + α ( α ( 68504986 + α ( α ( 20068023 + α ⁡ 28251722 − α ⁡ 27597594 + α ⁡ α ⁡ 227401 + 143868 ⁢ α − 8080506 ) − 7124277 8 ) ) − 34085604 ) ) − 1772332 ) ) ) ) θ 9 + 2 α 2 ( 900 + α ( α ( 227031 + α ( α ( 3860035 + α α ( 1446918 + α ( 8470466 + α ( α ⁡ 6780147 + α ⁡ 96634 ⁢ α − 1579311 − 12081405 ) ) ) − 5940428 ) ) − 1252830 ) ) − 22002 ) ) θ 10 + α 4 ( 2223 + α ( α ( 214140 + α ( α ( α ( 31 10570 + α ⁡ α ⁡ 5884380 + α ⁡ 356272 ⁢ α − 2400715 − 6582404 ) − 7509 ) − 548218 ) ) − 35082 ) ) θ 11 + 2 α 6 ( 408 + α ( α ( 9459 + α ( 33892 + α ( α ( 291642 + α ( 46571 ⁢ α − 194450 ) ) − 182296 ) ) ) − 4224 ) ) θ 12 + α 8 ( 18 + α ( 1176 + 5 α ( α ( 3978 + α ( 1402 ⁢ α − 4 011 ) ) − 1686 ) ) ) θ 13 + 6 α 10 ⁢ α ⁡ 42 + 11 ⁢ α − 3 ⁢ α − 12 ⁢ θ 14 + 3 1 − 2 α ⁢ α 12 ⁢ θ 15 H 23 = 804 − θ ⁡ θ ⁡ θ ⁡ 1156 + θ ⁡ 1248 + θ ⁡ 299 + 20 ⁢ θ − 2486 − 3807 + α 8 ⁢ θ − 1 ⁢ θ ( 1 + θ ( 1 4 + θ ) ) θ ⁡ θ ⁡ θ ⁡ θ ⁡ θ − 16 ⁢ θ − 149 − 554 − 275 − 30 − 1 + 2 α 7 ⁢ θ ( 1 + θ ⁡ 14 + θ ) ( θ θ ⁡ θ ⁡ θ ⁡ 1160 + θ ⁡ 208 + 29 ⁢ θ − 188 − 1159 − 412 − 22 ) + 2 α ( θ ( θ ( θ ( θ ( 6654 + θ ( 2635 + 269 ⁢ θ ) ) − 501 ) − 15453 ) − 9914 ) − 1672 ) + α 2 ( 5340 + θ ( 43002 − θ ( θ ( θ ( 436 60 + θ ⁡ 34092 + 59 ⁢ θ ⁡ 93 + θ ) − 57729 ) − 103425 ) ) ) + 2 α 3 ( θ ( θ ( θ ( θ ( 12288 + θ ( 515 95 + 6 θ ⁡ 2303 + 72 ⁢ θ ) ) − 90827 ) − 83552 ) − 24226 ) − 2000 ) + α 4 ( 1340 + θ ( 29000 + θ ⁡ 149089 − θ ⁡ θ ⁡ θ ⁡ 139800 + θ ⁡ 74231 + 5028 ⁢ θ + 57 ⁢ θ 2 − 68455 − 255200 ) ) + 2 α 5 ( θ ( θ ⁡ θ θ ⁡ θ ⁡ 48302 + θ ⁡ 48861 + θ ⁡ 6946 + 273 ⁢ θ − 65181 − 96842 − 36393 − 4198 ) − 72 ) + α 6 ( 4 + θ ( 910 − θ ( θ ( θ ( θ ( 28137 + θ ( 74673 + θ ( 14575 + θ ( 1679 + 1 7 θ ) ) ) ) − 98339 ) − 77899 ) − 17193 ) ) )

H 24 = 5 + θ + α ⁡ α − 6 − 8 θ + α θ ⁡ 6 + θ H 25 = − 8 ( α − 1 ) 6 ⁢ 2661 + α ⁡ α ⁡ 1413 + α ⁡ 4 α − 125 − 5949 + 4 ( − 1 + α − 1 + α 4 ( α ( 19 0524 + α ⁡ α ⁡ 367280 + α ⁡ α ⁡ 24028 + α ⁡ 36 ⁢ α − 1561 − 155498 − 383343 ) − 42 226 ) θ + ( α − 1 ) 3 ( 542789 + α ( − 3826760 + α ( 10880859 + α ( α ( 16843848 + α ( α ( 2543492 + α ⁡ α ⁡ 14667 + α − 260 ⁢ α − 296032 ) − 9182064 ) ) − 17527852 ) ) ) ) θ 2 + ( − 1 + α ) 2 ( α ( 7577789 + α ( α ( 82888389 + α ( α ( 115756930 + α ( α ( 26179168 + α ⁡ α ⁡ 439917 + α ⁡ 155 ⁢ α − 15107 − 5146093 ) − 70980498 ) ) ) − 121329016 ) ) − 3 4794805 ) − 583045 ) θ 3 + 2 α − 1 ( 71953 + α ( α ( 20507632 + α ( α ( 181963802 + α α ( 251356206 + α ( α ( 72172215 + α ( α ⁡ 2691103 + 2 α ⁡ 1391 ⁢ α − 77317 − 19155 772 ) ) − 165808443 ) ) − 260148457 ) ) − 81143287 ) ) − 2356214 ) ) θ 4 + 2 ( 71139 + α − 422088 + α ( α ( 47173488 + α ( α ( 456322989 + α ( α ( 677431203 + α ( α ( 233007 823 + α ⁡ α ⁡ 14421084 + α ⁡ 40549 ⁢ α − 1362117 − 75066439 ) − 477133971 ) ) − 6 73423632 ) ) − 197102912 ) ) ) − 3957246 ) ) θ 5 + ( 116953 + α ( − 2020445 + α ( 12592 236 + α ( − 28614206 + α ( − 33005086 + α ( 323127516 + α ( − 793739900 + α ( 109 5685018 + α ( α ( 605957938 + α ( α ⁡ 66950388 + α ⁡ 511865 ⁢ α − 9577987 − 2531 64292 ) ) − 984817488 ) ) ) ) ) ) ) ) ) θ 6 + ( 36051 + α ( − 864577 + α ( 8432202 + α ( − 427 36052 + α ( 118109504 + α ( − 160696218 + α ( 31350108 + α ( 228968398 + α ( α ( 327010206 + α ⁡ 3 α ⁡ 21555340 + α ⁡ 390807 ⁢ α − 4509065 − 179660908 ) − 3822 70002 ) ) ) ) ) ) ) ) ) θ 7 + 2 ( 2620 + α ( − 79738 + α ( 1031664 + α ( − 7301255 + α ( 30731 325 + α ( − 78689616 + α ( 120552833 + α ( α ( 47809929 + 2 α ( α ( α ( 1883697 + 2 α 83314 ⁢ α − 445741 ) − 2077371 ) − 3204635 ) ) − 105800363 ) ) ) ) ) ) ) ) θ 8 − 2 ( − 150 + α ( 5620 + α ( − 95841 + α ( 919817 + α ( − 5342812 + α ( 19341549 + α ( α ( 5941544 3 + 2 α ( α ⁡ 11948385 + α ⁡ α ⁡ 3198294 + α ⁡ 146581 ⁢ α − 1152321 − 5628698 − 2 3838393 ) ) − 43576432 ) ) ) ) ) ) ) θ 9 + α 2 ( 1800 + α ( α ( 464379 + α ( α ( 8678112 + α ( α ( 13058415 + α ( 2152525 − α ( 11149126 + α ( α ⁡ 906605 + 117293 ⁢ α − 6454530 ) ) ) − 15927118 ) ) ) ) − 2639949 ) ) − 44340 ) θ 10 + α 4 ( 2223 + α ( − 35625 + α ( 226210 + α ( α ( 513057 + α ( 1794227 + α ( 3 α ⁡ 1561962 + α ⁡ 91063 ⁢ α − 643493 − 488397 8 ) ) ) − 658356 ) ) ) ) θ 11 + 2 α 6 ( 408 + α ( − 4430 + α ( 12681 + α ( 15019 + α ( − 133295 + α ⁡ 240213 + 10 ⁢ α ⁡ 4161 ⁢ α − 16994 ) ) ) ) ) θ 12 + 6 α 8 ( 3 + α ( 173 + α ( α ( 2736 + α ( 11 11 ⁢ α − 2975 ) ) − 1193 ) ) ) θ 13 + α 10 ⁢ α ⁡ 240 + α ⁡ 83 ⁢ α − 169 − 72 ⁢ θ 14 + 3 − 5 α ⁢ α 12 ⁢ θ 15 .

Therefore, cross-ownership decreases the profits of the industry while increasing consumer surplus and social welfare.

Proof of Proposition 2.

We substitute Eq. (9) into O_i=(1-α)π_i+απ_j and obtain the following. First, ∂ O i P / ∂ β i P = α − 1 x 1 w H 27 + x i H 28 + H 29 β i + x j H 29 + H 28 β j / H 26 2 , where H 26 = 15 − 32 α + 16 α 2 H 27 = 4 α − 3 α 27 + 8 α 2 α − 5 − 4 H 28 = 16 + α 16 3 − α α − 49 H 29 = 4 α − 1 1 + α 13 + 16 α − 2 α .

Therefore, the equilibrium disclosure of R&D knowledge is β i P = − w + x j / x i < 0 . Additionally, the second-order conditions are ∂ 2 O i P / ∂ β i P 2 = α − 1 H 29 x i 2 / H 26 2 > 0 . Accordingly, the two private firms fully disclose their information with cross-ownership created in the first stage in a private duopoly, so β 1 P = β 2 P = 1 .

Proof of Lemma 3.

(i) First, because ∂ x 1 P / ∂ x 2 P = ∂ x 2 P / ∂ x 1 P = 4 − 2 α / 21 − 38 α + 16 α 2 > 0 , x 1 P increases with x 2 P , while x 2 P increases with x 1 P . Thus, the R&D levels of firms are strategic substitutes for a private duopoly.

According to Eq. (10), we obtain ∂ x i P / ∂ α = 2 w 4 α − 11 H 30 / H 31 2 > 0 , where H₃₀=4α-5 and H₃₁=17+4α(4α-9). Thus, in a private duopoly, the R&D levels of both firms increase with cross-ownership, the R&D levels of the firms are strategic substitutes, and the R&D investments of the two firms are the same.
We substitute β 1 P = β 2 P = 1 and Eq. (10) into Eq. (9), and obtain ∂ q i M / ∂ α = 3 w 9 − 8 α / H 31 2 > 0 . Therefore, both two firms’ outputs increase with cross-ownership.

Proof of Lemma 4.

(i) We substitute Eq. (9), β 1 P = β 2 P = 1 and Eqs. (10) into Eq. (1), and obtain ∂ π i P / ∂ α = w 2 2 α − 1 H 30 H 32 / H 31 3 > 0 , where H₃₂=105 + 16α(2α-7). Therefore, cross-ownership increases the profits of the two private firms.

We substitute Eq. (9), β 1 P = β 2 P = 1 and Eq. (10) into Eq. (1), respectively, and obtain ∂ P S P / ∂ α = 2 w 2 2 α − 1 H 30 H 32 / H 31 3 > 0 , ∂ C S P / ∂ α = 12 w 2 1 − α H 30 8 α − 9 / H 31 3 > 0 and ∂ S W P / ∂ α = 2 w 2 H 30 8 α 53 − 38 α + 8 α 2 − 159 / H 31 3 > 0 . Therefore, in a private duopoly, cross-ownership increases the profits of the industry, consumer surplus and social welfare.

Proof of Proposition 3.

(ii) First, because x 1 M − x 1 P = w 2 α − 2 + H 16 H 31 / H 7 H 31 > 0 and x 2 M − x 2 P = 2 w α − 2 − H 2 H 31 H 33 / H 31 < 0 , x 1 M > x 1 P and x 2 M < x 2 P , where H₃₃=α³θ(23+(18-25θ)θ)-2(2+θ)²+α⁴(θ-1)θ(1+θ(14+θ))+α(16+θ(33+8θ))+α²(θ(θ(8θ-9)-47)-8). Therefore, the R&D level of firm 1 is higher than that of firm 1 in a private duopoly, while the R&D level of firm 2 in a mixed duopoly is lower than that of firm 2 in a private duopoly.

Second, because q 1 M − q 1 P = w α − 1 H 1 H 2 α + 3 H 2 H 31 − H 30 / H 7 H 31 > 0 and q 2 M − q 2 P = w H 2 H 1 H 5 H 31 − H 34 / H 7 H 31 > 0 , q 1 M > q 1 P and q 2 M > q 2 P , where H₃₄=5-9α+4α². Therefore, both firm 1 and firm 2 have higher output in a mixed duopoly than in a private duopoly.

Eventually, because π 1 M − π 1 P = w 2 H 31 2 H 36 − 2 H 35 / 2 H 7 2 H 31 2 < 0 and π 2 M − π 2 P = − w 2 H 35 + H 31 2 H 37 / H 31 2 H 7 2 > 0 , π 1 M < π 1 P and π 2 M > π 2 P , where.

H 35 = α − 2 ⁢ α ⁡ 63 + 8 α ⁡ 2 α − 7 − 21 H 36 = α ⁡ 1104 + α ⁡ 142 + α ⁡ 3 α − 56 − ( α − 1 ) 4 ⁢ 1303 + 2 ( α − 1 ) 3 ( α ( 584 + α ( α ( 3639 + α ⁡ 491 + α ⁡ 4 α − 115 ) − 4171 ) ) − 1704 ) θ − ( α − 1 ) 2 ( α ( 37276 + α ( α ( 82134 + α ⁡ α ⁡ 16642 + α ⁡ 2457 − 328 ⁢ α + 6 α 2 − 61837 ) − 71321 ) ) − 4341 ) θ 2 − 2 α − 1 502 + α ( α ( 73779 + α ( α ( 202938 + α ( α ⁡ 60905 + α ⁡ 2 α ⁡ 39 ⁢ α − 626 − 8727 − 147486 ) ) − 168733 ) ) − 11984 ) ) θ 3 + ( 2 α ( 482 + α ( 16878 − α ( 122067 + α ( α ( 462354 + α ⁡ α ⁡ 191245 + α ⁡ α ⁡ 4966 + 413 ⁢ α − 52542 − 382771 ) − 328577 ) ) ) − 411 ) ) θ 4 + 2 ( α ( 1520 + α ( α ( 6108 + α ( 58501 + α ( α ( 272973 + 2 α ( α ( 44208 + α ⁡ 734 ⁢ α − 9815 ) − 103192 ) ) − 195482 ) ) ) − 7386 ) ) − 105 ) θ 5 + ( − 25 + 2 α ( 250 + α ( α ( 9222 + α ( α ( 3431 + α ( 42491 + α ( α ⁡ 49140 + α ⁡ 2739 ⁢ α − 17618 − 693 22 ) ) ) − 18184 ) ) − 2134 ) ) ) θ 6 + 2 α 2 ( − 20 + α ( 300 + α ( − 1704 + α ( 4330 + α ( α ( α ⁡ 4131 + 10 ⁢ α ⁡ 37 ⁢ α − 225 − 996 ) − 4159 ) ) ) ) ) θ 7 + α 4 ( α ( 60 + α ( α ( 418 + α ( 2 α ⁡ 13 ⁢ α − 50 − 193 ) ) − 216 ) ) − 6 ) θ 8 + 2 α 6 ⁢ 4 + α ⁡ − 20 + α ⁡ 17 + 2 α ⁢ θ 9 − α 8 ⁢ θ 10

H 37 = α θ − 1 ) 2 ( 8 α − 15 ⁢ 7 − α ( α − 1 ) 4 + 2 ( α − 1 ) 3 ( 708 + α ( α ( 2749 + α ( 11 ⁢ α − 35 5 ) ) − 3537 ) ) θ − ( α − 1 ) 2 ( 882 + α ( α ( 28166 + α ( α ⁡ 11627 + α ⁡ 13 ⁢ α − 906 − 30 657 ) ) − 8901 ) ) θ 2 − α − 1 ( α ( 3760 + α ( α ( 57685 + α ( α ( 42401 − 9752 α + 370 α 2 ) − 72463 ) ) − 21777 ) ) − 240 ) θ 3 + ( − 24 + α ( 528 + α ( α ( 21486 + α ( α ( 71966 + α ⁡ 20024 − 2739 ⁢ α ⁢ α − 53900 ) − 52569 ) ) ) − 4764 ) ) θ 4 − 2 α 2 ( 48 + α ( α ( 1575 + α ⁡ α ⁡ 3975 + α ⁡ 734 ⁢ α − 2741 − 3141 ) − 432 ) ) θ 5 + α 4 ( 126 + α ( α ( 1992 + α ( 413 ⁢ α − 1628 ) ) − 855 ) ) θ 6 + 3 α 6 ⁢ 13 − 46 ⁢ α + 26 ⁢ α 2 ⁢ θ 7 + 3 α 8 ⁢ θ 8 .

Therefore, the profit of firm 1 in a mixed duopoly is lower than that of firm 1 in a private duopoly, and the profit of firm 2 in a mixed duopoly is also lower than that of firm 2 in a private duopoly.

Because P S M − P S P = w 2 H 31 2 H 38 − 4 H 35 / H 31 2 2 H 7 2 < 0 , C S M − C S P = w 2 H 2 2 H 24 2 H 1 2 H 31 2 − 4 H 34 2 / 2 H 7 2 H 31 2 > 0 and S W M − S W P = w 2 H 40 H 31 2 − H 39 / 2 H 7 2 H 31 2 > 0 , P S M < P S P , C S M > C S P and S W M > S W P , where

H 38 = 377 − 2 α 10 ⁢ θ 2 ( 1 + θ ⁡ 14 + θ ⁢ ) 2 ⁢ 3 + θ ⁡ θ ⁡ 3 θ − 2 ⁢ θ − 26 − 6 + θ ( 6240 + θ ( 6105 − θ θ ⁡ 363 + 5 θ ⁡ 42 + 5 θ − 1484 ) ) + 4 α 9 ⁢ θ ⁡ 1 + θ ⁡ 14 + θ ( 2 + θ ( 57 + θ ( θ ( θ ( θ ( 73 ⁢ θ − 155 ) − 1356 ) − 1150 ) − 159 ) ) ) + 4 α ( θ ( θ ( θ ⁡ θ ⁡ θ ⁡ 736 + 125 ⁢ θ − 263 − 9125 − 182 38 ) − 9349 ) − 741 ) + α 2 ( 7960 + 2 θ ( 49075 + θ ( 145441 − 2 θ ( θ ( θ ( 2997 + 5 θ ( 211 + 2 θ ) ) − 15262 ) − 67023 ) ) ) ) + 4 α 3 ( θ ( θ ( θ ( θ ⁡ θ ⁡ 3 θ ⁡ 1417 + 50 ⁢ θ − 4142 − 102646 − 213498 ) − 146194 ) − 35477 ) − 2421 ) + 2 α 4 ( 2609 + θ ( 58139 + θ ( 335256 + θ ( 724 567 + θ ⁡ 586920 − θ ⁡ θ ⁡ 11818 + 3 θ ⁡ 536 + θ − 130160 ) ) ) ) + 4 α 5 ( θ ( θ ( θ ( θ ( θ ( θ ( θ ⁡ 1859 + 15 ⁢ θ − 11750 ) − 193182 ) − 445253 ) − 353672 ) − 110965 ) − 12189 ) − 15 1 ) + 2 α 6 ( − 184 + θ ( 3009 + θ ( 76764 + θ ( 395898 + θ ( 767506 + θ ( 555003 + 2 θ ( 52 816 + θ ⁡ θ ⁡ 2 θ − 117 − 1379 ) ) ) ) ) ) ) − 4 α 7 ( − 17 + θ ( − 405 + θ ( 4196 + θ ( 57176 + θ 185196 + θ ⁡ 217265 + θ ⁡ 77780 + θ ⁡ 1578 + θ ⁡ 10 ⁢ θ − 571 ) ) ) ) − α 8 ( 3 + θ ( 254 + θ ( 2989 + θ ( θ ( θ ⁡ θ ⁡ θ ⁡ θ ⁡ 4735 + θ ⁡ 44 + θ − 17494 − 227182 − 364170 − 17795 4 ) − 20970 ) ) ) ) H 39 = 4 67 − 237 ⁢ α

H 40 = 2 ( α − 1 ) 4 ⁢ 1701 − α ⁡ 1608 + α ⁡ α − 12 ⁢ α − 120 + 2 ( α − 1 ) 3 ( α ( 23474 + α ( α ( 107 60 + α ⁡ α ⁡ 3 α − 62 − 429 ) − 28625 ) ) − 4825 ) θ − ( α − 1 ) 2 ( α ( 85376 + α ( α ( 39725 4 + α ⁡ α ⁡ 51706 + α ⁡ α ⁡ 5 α − 214 − 692 − 235962 ) ) − 290211 ) − 7506 ) θ 2 − 2 ( α − 1 ) 2 ( α ( 20427 + α ( α ( 340345 + 2 α ( α ⁡ 88070 + α ⁡ α ⁡ 29 ⁢ α − 25 − 14384 − 18584 0 ) ) − 134632 ) ) − 866 ) θ 3 − ( 347 + α ( 1964 + α ( α ( 539372 + α ( α ( 2789188 + α ( 4 α 334780 + α ⁡ α ⁡ 7798 + 57 ⁢ α − 85925 − 2598421 ) ) − 1682070 ) ) − 77207 ) ) ) θ 4 + 2 ( 2 α ( 728 + α ( α ( α ( 90507 + α ( α ( 431919 + α ( α ⁡ 159296 + α ⁡ 1789 ⁢ α − 32320 − 360197 ) ) − 280987 ) ) − 7945 ) − 105 ) − 2717 ) ) θ 5 + ( α ( 500 + α ( α ( 16508 + α ( 2 α ( α ⁡ 168324 + α ⁡ α ⁡ 191195 + α ⁡ 9577 ⁢ α − 69816 − 253998 − 43082 ) − 17389 ) ) − 4204 ) ) − 25 ) θ 6 + 2 α 2 ( α ( 300 + α ( α ( 3313 + 2 α ( 134 + α ( α ( 11227 + α ( 1789 ⁢ α − 7535 ) ) − 6551 ) ) ) − 1588 ) ) − 20 ) θ 7 + α 4 ( − 6 + α ( 60 + α ( − 435 + 2 α ( 940 + α ( − 16 05 + 2 442 − 57 ⁢ α ⁢ α ) ) ) ) ) θ 8 − 2 α 6 ⁢ α ⁡ 20 + α ⁡ 17 + α ⁡ 58 ⁢ α − 117 − 4 ⁢ θ 9 − α 8 ( 1 + 5 α 2 ) θ 10 .

Therefore, the industry profit in a mixed duopoly is lower than that in a private duopoly, while consumer surplus and social welfare in a mixed duopoly are higher than those in a private duopoly.

Proof of Proposition 4.

(i) First, through calculations, we obtain ∂ O 2 U 1 ∂ β 2 U 1 = 4 α − 1 x 2 w 2 + θ + x 1 3 + θ β 1 − 1 + x 2 3 + θ − β 2 α − 11 − 4 θ 2 = 4 α − 1 x 2 11 − α + 4 θ × q 2 U 1 < 0 , so β 2 U 1 = 0 . Substituting β 2 U 1 = 0 into ∂ O 1 U 1 ∂ β 1 U 1 , we find that ∂ O 1 U 1 ∂ β 0.5 U 1 > 0 , so β 1 U 1 = 1 . Accordingly, in the case of unilateral shareholding, the semi-public firm fully discloses its R&D knowledge, while the private firm does not disclose its R&D knowledge β 1 U 1 = 1 , β 2 U 1 = 0 .

Because ∂ x 1 U 1 ∂ x 2 U 1 = 28 − α 2 2 + θ + α 5 + 2 θ 4 + 3 θ − θ 13 + θ 22 + 5 θ 2 + θ 31 + α 2 − 2 α 7 + 3 θ + θ 27 + 5 θ > 0 and ∂ x 2 U 1 ∂ x 1 U 1 = 4 2 + θ 3 + θ α − 17 − 6 θ α − 5 − 2 θ > 0 , in the case of unilateral shareholding, the R&D levels of firms are strategic complements, and the R&D level chosen by one firm increases with that chosen by the other.

Additionally, because x 1 U 1 − x 2 U 1 = w H 44 / 2 + θ H 42 > 0 , x 1 U 1 > x 2 U 1 , where H₄₄= 253 + α³(1 + θ)-α²(17 + θ(33 + 10θ))-θ(115 + θ(339 + θ(151 + 20θ))) + α(43 + θ(243 + θ(163 + 29θ))). Therefore, the semi-public firm invests more in R&D than do the private firm.

Because ∂ x 1 U 1 ∂ α = 2 w H 45 2 + θ H 42 2 > 0 , ∂ x 2 U 1 ∂ α = 8 w 3 + θ H 46 H 42 2 > 0 , both the semi-public and private firms’ R&D levels increase with cross-ownership, where H 45 = 46706 + α 4 4 + θ − α 3 154 + 5 θ 19 + 3 θ − α 11 + 4 θ ( 1498 + θ ( 1499 + θ ( 511 + 60 θ ) ) ) + α 2 2322 + θ 2295 + θ 769 + 88 θ + θ ( 81184 + θ ( 56791 + 8 θ ( 2502 + θ 445 + 32 θ ) ) ) H 46 = 847 − α 3 − 5 α 5 + 2 θ 11 + 4 θ + α 2 29 + 11 θ θ 985 + 6 θ 63 + 8 θ

Because ∂ S W U 1 ∂ α = w 2 H 47 2 + θ 2 H 42 3 < 0 , cross-ownership decreases social welfare, where H 47 = 4 11 − α 3 545 + α α 1125 + α 4 α − 109 − 4481 − 4 α − 11 ( 941965 + α ( − 21146 79 + α 716331 + α − 105487 + 2 α 3972 + α − 147 + 2 α ) ) θ + ( 102246919 + α ( α 50473599 + α α 622205 + α 493 − 2 α α − 26292 − 7712842 ) − 150320528 ) θ 2 + ( 120151745 + α ( α 33925356 + α α 236873 − 6306 α + 50 α 2 − 4037300 − 1265 72290 ) ) θ 3 + ( 83463409 + α ( − 64751466 + α ( 13359798 + α ( − 1158248 + 43569 α − 5 34 α 2 ) ) ) ) θ 4 + 2 18553065 + α α 1560576 + α 1575 α − 87538 − 10452350 θ 5 − 8 ( α 524855 + 9 α 153 α − 5615 − 1347073 ) θ 6 + 16 124579 + α 1410 α − 30179 θ 7 + 64 3349 − 382 α θ 8 + 10240 θ 9 .

Proof of Proposition 5.

(i) First, through calculations, we obtain

∂ O 1 U 2 ∂ β 1 U 2 = x 1 − w H 54 + x 1 − H 55 + H 56 β 1 + x 2 H 56 − H 55 β 2 ) / H 48 2 ∂ O 2 U 2 ∂ β 2 U 2 = − α θ − 1 x 2 w H 57 + + x 1 H 58 − H 59 β 1 + x 2 H 59 + H 58 β 2 / H 48 2 ,

where

H 54 = θ ( 13 + 78 α + 22 θ − 2 α 37 + 27 α θ + 5 + α 87 α − 40 θ 2 − α 2 ( α 12 + α − 3 ) θ 3 + α 4 θ 4 ) − 28 H 55 = α θ − 1 θ θ − 3 + α 30 + θ 1 + θ + α α θ − 1 − 3 θ − 13 − 14 H 56 = 4 α θ − 3 − θ θ 8 + 5 θ + α 22 + θ 2 α θ − 20 − 3 α − 14 H 57 = θ α 37 + θ 24 + α 3 α θ − 52 − 4 − 8 H 58 = 4 1 + α θ 7 + θ 8 + α α θ − 17 H 59 = 12 + θ 4 + α θ 8 + α α θ − 16 − 9 .

Therefore, the equilibrium disclosure of R&D knowledge is β 1 U 2 = − w + x 2 /x₁<0 and β 2 U 2 = − w + x 1 / x 2 < 0 . Additionally, the second-order conditions are ∂ 2 O 1 U 2 ∂ β 1 U 2 2 = 4 α − 1 θ − 3 H 56 x 1 2 H 48 2 > 0 and ∂ 2 O 2 U 2 ∂ β 2 U 2 2 = − 4 α θ − 1 H 58 x 2 2 H 48 2 > 0 . Accordingly, both semi-public and private firms fully disclose their information with cross-ownership, so β 1 U 2 = β 2 U 2 = 1 .

Because ∂ x 1 U 2 / ∂ x 2 U 2 = ( θ ( 279 α − 85 + 262 − 437 α α − 37 θ + ( α 56 + α 258 α − 217 − 5 θ 2 − 2 α 2 2 + α 17 α − 12 θ 3 + α 4 1 + α θ 4 ) − 62 ) / H 48 2 > 0 and ∂ x 2 U 2 / ∂ x 1 U 2 = − 3 35 + 4 θ 6 + θ + α θ θ α 96 + θ 8 + α α θ − 22 − 40 − 106 / H 48 2 > 0 , in the case of unilateral shareholding, the R&D levels of firms are strategic complements, and the R&D level chosen by one firm increases with that chosen by the other.

Because x 1 U 2 − x 2 U 2 = w H 60 / H 53 − 46 > 0 , x 1 U 2 > x 2 U 2 , where H₆₀=θ(13 + 96α³θ² + 2α⁵θ⁴ + 5θ(5 + θ)-α⁴θ³(35 + θ)+α²θ(θ(73 + 4θ)-169)-2α(θ(35 + 22θ)-72))). Therefore, the semi-public firm invests more in R&D than do the private firm.

First, because ∂ x 1 U 2 ∂ α = 3 w θ H 61 H 53 2 > 0 and ∂ x 2 U 2 ∂ α = 2 w θ H 62 H 53 2 > 0 , both semi-public and private firms’ R&D levels increase with cross-ownership, where

H 61 = ( 1529 + θ ( 1153 − 10326 α + α ⁡ 27401 ⁢ α − 3282 − 805 ⁢ θ − ( 893 + α ( α ( 497 + 366 88 ⁢ α ) − 5572 ) ) θ 2 + 2 α ⁡ 1598 + α ⁡ α ⁡ 4832 + 13187 ⁢ α − 6073 − 122 ⁢ θ 3 − 2 ( 10 + α ⁡ α ⁡ 1711 + α ⁡ α ⁡ 5334 + 5029 ⁢ α − 5278 − 224 ) θ 4 + α 2 ( α ( 988 + α ( 3 α ( 1406 + 631 ⁢ α ) − 3397 ) ) − 104 ) θ 5 − α 4 ⁢ 69 + α ⁡ α ⁡ 721 + 104 ⁢ α − 440 ⁢ θ 6 − α 6 ( 20 + ( α − 32 ) α ) θ 7 + α 8 ⁢ θ 8 ) H 62 = ( 682 + θ ( 1459 − α 8 ⁢ θ − 16 ⁢ θ 7 − 8 α 7 ⁢ θ 6 ⁢ 59 + 4 θ + α 6 ⁢ θ 5 ⁢ 5243 + θ ⁡ 821 + 20 ⁢ θ − 4 α 5 ⁢ θ 4 ⁢ 4441 + θ ⁡ 2153 + 96 ⁢ θ + α 4 ⁢ θ 3 ⁢ 30676 + θ ⁡ 23288 + 5115 ⁢ θ + 51 ⁢ θ 2 + θ ( 1033 + θ ⁡ 233 − 2 θ ⁡ 11 + 5 θ ) − 4 α ⁡ 2 + θ ⁢ 688 + θ ⁡ 771 + θ ⁡ 249 + 14 ⁢ θ − 8 α 3 ⁢ θ 2 ( 3841 + 4 θ ⁡ 955 + θ ⁡ 349 + 42 ⁢ θ ) + α 2 ⁢ θ ⁡ 17861 + θ ⁡ 22507 + 4 θ ⁡ 2640 + θ ⁡ 517 + 32 ⁢ θ ) .

Because ∂ S W U 2 ∂ α = w 2 θ H 63 H 53 3 > 0 , cross-ownership increases social welfare, where H 63 = − 46508 + θ ( 809825 α − 345799 + 3769477 − 5419905 α α − 430554 θ + ( α 3318231 + α 19600853 α − 17313735 − 77309 ) θ 2 + ( 159425 − α ( 116167 + α 10287948 + α 43701401 α − 44300815 ) ) θ 3 + ( 119439 + α ( α ( 2555307 + α 16460523 + α 63997821 α − 70238377 ) − 1318247 ) ) θ 4 + ( 36211 − α ( 645231 + α α 7673237 + α 14707263 + α 63516316 α − 72805713 − 3976725 ) ) θ 5 + ( 5240 + α ( α ( 1235262 + α ( α ( 9893375 + α ( 8082675 + α ( 43232263 α − 51023 273 ) ) ) − 5598874 ) ) − 127756 ) ) θ 6 + ( 300 − α ( 9440 + α ( α ( 982806 + α ( α ( 59471 85 + α 3834771 + α 20059620 α − 24667523 ) − 3747949 ) ) − 133422 ) ) ) θ 7 + α 2 ( 1800 + α ( α ( 283011 + α ( α ( 1465661 + α ( 1881549 + α ( 6132589 α − 806576 1 ) ) ) − 1056591 ) ) − 35700 ) ) θ 8 + α 4 ( 2223 − α ( 27291 + α ( α ( 126443 + α ( 523524 + α 1132867 α − 1585193 ) ) − 117331 ) ) ) θ 9 + 2 α 6 ( 408 + α ( α ( 4143 + α ( 27321 + α 54375 α − 76003 ) ) − 3368 ) ) θ 10 + α 8 ( 18 + α ( 984 + α ( 13 612 − 407 α α − 536 7 ) ) ) θ 11 + α 10 α 204 + α 107 α − 121 − 72 θ 12 + 3 − 5 α α 12 θ 13 ) ) .

Proof of Proposition 6.

Profit maximisation at the different stages of the game allows us to compute the equilibrium values of output, R&D investments and profits in the symmetric subgames (I/I) and (NI/NI) as well as in the asymmetric subgame (I/NI). Under the condition that both stability and R&D conditions are satisfied in both symmetric and asymmetric subgames, we obtain the following equilibrium values:

Table 2:

The equilibrium values in the symmetric subgame (I/I).

	The symmetric subgame (I/I)
q	q 1 * I / I = α − 1 ( w α + 3 H 2 + x 1 4 H 2 + α − H 2 β 1 + x 2 α − H 2 + 4 H 2 β 2 / H 1 q 2 * I / I = w H 4 + x 1 1 + α θ − 1 H 2 + H 3 β 1 + x 2 H 3 + 1 + α θ − 1 H 2 β 2 / H 1
β	β 1 * I / I = β 2 * I / I = 1
x	x 1 * I / I = w 2 α 5 H 8 − 59 + θ θ 21 + 5 θ − 3 + α H 9 / H 7 x 2 * I / I = ( 2 w H 2 ( 2 2 + θ 2 − α H 10 ) ) ) / H 7
O	O 1 * I / I = ( w 2 ( − 2 ( α − 1 ) 4 α 768 + α 56 + α − 20 α − 1701 + ( α − 1 ) 3 ( α ( 35996 + α α 7008 + α 1529 + α 9 α − 260 − 37049 ) − 4945 ) θ − ( α − 1 ) 2 ( α ( 349 98 + α ( α 251642 + α α 5570 + α 5752 + α 13 α − 584 − 120186 − 17 6633 ) ) − 1264 ) θ 2 + α − 1 ( 1433 + α ( α ( α ( 439023 + α ( α ( 608080 + α ( α ( α 9040 + α 6 α − 547 − 15158 ) − 158584 ) ) − 804810 ) ) − 73697 ) − 4818 ) ) θ 3 + ( α ( 16426 + α ( α ( 91442 + α ( 369982 + α ( α ( 1354291 + 2 α ( α ( 41200 + α 14528 + 7 α 13 α − 448 ) − 329222 ) ) − 1197002 ) ) ) − 81015 ) ) − 1075 ) θ 4 + ( α ( 6251 + α ( α ( 232486 + α ( α ( 226696 + α ( 437741 + 2 α ( α ( 162383 + α α 783 α − 7355 − 12883 ) − 342820 ) ) ) − 447538 ) ) − 55379 ) ) − 274 ) θ 5 + ( α ( 838 + α ( α ( 75324 + α ( 2 α ( 271275 + α ( α ( 50180 + α ( 81165 + α ( α 4137 + 1271 α − 48771 ) ) ) − 253990 ) ) − 276071 ) ) − 11148 ) ) − 25 ) θ 6 + α ( 25 − α ( 604 + α ( α ( 40599 + 2 α ( α ( 129608 + α ( α ( 52439 + α ( 7396 + α 1271 α − 9702 ) ) − 130484 ) ) − 68106 ) ) − 6804 ) ) ) θ 7 + α 3 ( 40 + α ( α ( 5502 + α 2 α 17079 + α α 7302 + 712 − 783 α α − 17999 − 17949 ) ) − 838 ) θ 8 + α 5 6 + α 167 + α α 2565 + α 2 263 − 91 α α − 1519 − 1286 θ 9 + α 7 α 107 + α 107 − 6 α α − 264 − 8 θ 10 + α 9 1 + 5 α θ 11 ) ) / 2 H 7 2 O 2 * I / I = ( w 2 ( − 16 α − 15 α − 7 ( α − 1 ) 5 − ( α − 1 ) 4 ( α ( 18811 + 3 α ( α ( 766 + ( α − 4 4 ) α ) − 4652 ) ) − 2832 ) θ + 2 ( α − 1 ) 3 ( α ( 10029 + α ( α ( 53486 + α ( α ( 3053 + 4 α − 45 α ) − 21604 ) ) − 45402 ) ) − 882 ) θ 2 − ( α − 1 ) 2 ( α ( 6707 + α ( α ( 212 129 + α α 229661 + α α 8381 − 424 α + 6 α 2 − 66874 − 340208 ) − 48 242 ) ) − 480 ) θ 3 − 2 α − 1 ( 24 + α ( α ( 1662 + α ( α ( 89610 + α ( α ( 237249 + α α 27708 + α 91 α − 2924 − 118242 ) − 225214 ) ) − 9446 ) − 506 ) ) ) θ 4 + α ( 2 α ( 1922 + α ( 1094 + α ( α ( 66036 + α ( 10558 + α ( α ( 130570 − 3 α ( 18 909 + α 261 α − 3668 ) ) − 127405 ) ) ) − 36016 ) ) ) − 507 ) θ 5 + 2 α ( α ( 1544 + α ( α ( 13194 + α ( 25204 + α ( α ( 123594 + α ( α 1005 + 6069 − 1271 α α − 55695 ) ) − 105410 ) ) ) − 8130 ) ) − 105 ) θ 6 + α ( 2 α ( 250 + α ( α ( 9318 + α ( α ( 5444 + α 38930 + α α 39868 + α 1271 α − 11494 − 62546 ) − 18892 ) ) − 2134 ) ) − 25 ) θ 7 + 2 α 3 ( α ( 300 + α ( α ( 4204 + α ( α ( α ( 8186 + α ( 783 α − 4447 ) ) − 4200 ) − 3100 ) ) − 1704 ) ) − 20 ) θ 8 + α 5 ( α ( 60 + α ( α ( 340 + α ( 173 + 2 α 91 α − 272 ) ) − 216 ) ) − 6 ) θ 9 + 2 α 7 4 + α α 17 + α 3 α − 1 − 20 θ 10 − α 9 θ 11 ) ) / 2 H 7 2

Table 3:

The equilibrium values in the symmetric subgame (I/NI).

	The symmetric subgame (I/NI)
q	q 1 * I / N I = α − 1 w α + 3 H 2 + x 1 4 H 2 + α − H 2 β 1 / H 1 q 2 * I / N I = w H 4 + x 1 1 + α θ − 1 H 2 + H 3 β 1 / H 1
β	β 1 * I / N I = 1
x	x 1 * I / N I = w H 64 / H 65
O	O 1 * I / N I = − w 2 H 2 H 64 / 2 H 65 O 2 * I / N I = ( w 2 ( − 16 ( α − 11 ) 2 ( α − 1 ) 5 − ( α − 1 ) 4 ( α ( 20923 + α ( 2362 + 3 α − 44 α − 15876 ) ) − 3344 ) θ + 2 ( α − 1 ) 3 ( α ( 11597 + α ( α ( 59558 + α ( α ( 3205 + 2 α 2 α − 91 ) − 24688 ) ) − 49884 ) ) − 1074 ) θ 2 − ( α − 1 ) 2 ( α ( 8387 + α ( α ( 230209 + α ( α 256317 + α α 8909 + 6 α − 72 α − 77130 − 369 176 ) ) − 55762 ) ) − 608 ) θ 3 − 2 α − 1 ( 32 + α ( α ( 2838 + α ( α ( 94026 + α α 251555 + α α 32192 − 3120 α + 93 α 2 − 131186 − 233322 ) − 12390 ) ) − 698 ) ) θ 4 + α ( 2 α ( 2066 + α ( 1014 − α ( 38672 + α ( α ( 10366 + α 109801 + α α 60887 + α 835 α − 12896 − 127506 ) − 77348 ) ) ) ) − 539 ) θ 5 + 2 α ( α ( 1552 + α ( α ( 12282 + α ( 29156 + α ( α ( 141090 + α ( α 10787 + 9 515 − 173 α α − 74425 ) ) − 115342 ) ) ) − 8074 ) ) − 105 ) θ 6 + α ( 2 α ( 250 + α ( α ( 9254 + α ( α ( 5572 + α ( 37250 + α ( α ( 41988 + α ( 19 51 α − 13930 ) ) − 61574 ) ) ) − 18612 ) ) − 2134 ) ) − 25 ) θ 7 + 2 α 3 ( α ( 300 + α ( α 4324 + α α α 5022 + α 497 α − 2829 − 1536 − 4052 − 1704 ) ) − 20 ) θ 8 + α 5 α 60 + α α 404 + α 78 α 2 − 99 − 232 α − 216 − 6 θ 9 + 2 α 7 4 + α α 17 + α + α 2 − 20 θ 10 − α 9 θ 11 ) ) ) / 2 H 65 2

where H 64 = 2 α 5 θ − 1 θ 1 + θ 14 + θ − 59 + θ θ 21 + 5 θ − 3 + 2 α ( 56 + θ ( 91 − θ 47 + 25 θ ) ) + 2 α 2 θ θ 2 θ 40 + θ − 23 − 149 − 23 − 2 α 3 ( 4 + θ ( θ θ 57 + 10 θ − 99 − 44 ) ) − α 4 θ θ 54 + θ 28 + θ − 17 θ − 33 − 1 H 65 = α 5 θ 3 + θ 12 + θ 1 + θ 14 + θ − 2 + θ 31 + θ 27 + 5 θ + α ( 152 + θ 475 + 6 θ 57 + 11 θ ) − 2 α 2 60 + θ 381 + 2 θ 256 + θ 82 + θ + 2 α 3 16 + θ 227 + θ 591 + 2 θ 172 + 7 θ + α 4 ( θ ( θ ( θ ( θ − 58 θ − 58 4 ) − 520 ) − 85 ) − 2 ) .

Table 4:

The equilibrium values in the symmetric subgame (NI/I).

	The symmetric subgame (NI/I)
q	q 1 * N I / I = α − 1 w α + 3 H 2 + x 2 α − H 2 + 4 H 2 β 2 / H 1 q 2 * N I / I = w H 4 + x 2 H 3 + 1 + α θ − 1 H 2 β 2 / H 1
β	β 2 * N I / I = 0
x	x 2 * N I / I = ( w ( 4 2 + θ 3 + θ + α 4 θ 3 + θ 1 + θ 14 + θ − α 3 θ 69 + θ 142 + 29 θ − α 48 + θ 119 + 44 θ + α 2 24 + θ 165 + θ 143 + 8 θ ) ) / H 66
O	O 1 * N I / I = ( w 2 ( − ( α − 1 ) 4 α 620 + α 218 + α − 28 α − 4091 + ( α − 1 ) 3 ( α ( 2 7061 + α α α 1337 + α 2 α − 107 − 1546 − 21168 ) − 3051 ) θ + ( α − 1 ) 2 ( α ( α ( 65476 + α ( α 22595 + α 10512 + α 75 α − 2122 − 8 9093 ) ) − 5366 ) − 1277 ) θ 2 + α − 1 ( 1837 + α ( α ( 41890 + α ( 11138 + α α 71803 + 4 α 1002 + 5 α 46 α − 601 − 102423 ) ) − 17041 ) ) θ 3 + ( α ( 9334 + α ( α ( 160048 + α ( α ( 104477 + α ( 7300 + α ( α ( 3593 α − 6935 ) − 12290 ) ) ) − 208548 ) ) − 56375 ) ) − 600 ) θ 4 + ( α ( 1288 + α ( α 51128 + α ( α 161634 + α α 21020 + 49 63 − 22 α α − 99316 − 126554 ) ) − 11161 ) ) − 64 ) θ 5 − α 2 ( 112 + α ( α ( 8864 + α ( α ( 27745 + α α 1962 + 403 α − 14439 ) − 23068 ) ) − 1596 ) ) θ 6 − α 4 ( 36 + α ( α ( 11 67 + 2 α 3 α 59 + 6 α − 608 ) − 370 ) ) θ 7 − α 6 ( 4 + α ( α 14 + α − 20 ) ) θ 8 ) ) / 2 H 66 2 O 2 * N I / I = ( w 2 α − 1 ( 2 α 3 θ 23 + 29 − 12 θ θ − 4 2 + θ 2 + α ( 32 + 74 θ + 24 θ 2 ) + α 4 θ − 2 θ 1 + θ 14 + θ + α 2 θ − 8 2 + θ 13 + 8 θ ) ) / 2 H 66 2

where H 66 = 5 + 2 θ 17 + 6 θ + α 4 1 + θ 10 + θ 1 + θ 14 + θ + 2 α 2 5 + 4 θ ( 13 + θ 48 + θ ) − 4 α 48 + θ 97 + 30 θ − 4 α 3 6 + θ 71 + θ 106 + 9 θ ) .

Table 5:

The equilibrium values in the symmetric subgame (NI/NI).

	The symmetric subgame (NI/NI)
q	q 1 * N I / N I = w α − 1 α + 3 H 2 / H 1 q 2 * N I / N I = w H 4 / H 1
O	O 1 * N I / N I = − w 2 H 64 / 2 H 1 2 O 2 * N I / N I = w 2 α − 1 α H 10 − 2 2 + θ 2 / H 1 2

We now examine the first stage of the game, in which firms choose whether to invest in R&D. Making use of the firms’ objective function, we can build on the payoff matrix summarised in Table 6.

Table 6:

The investment-decision game (payoff matrix).

Firm i ↓ firm j →	I	NI
I	O i * I / I , O j * I / I	O i * I / N I , O j * I / N I
NI	O i * N I / I , O j * N I / I	O i * N I / N I , O j * N I / N I

To derive all of the possible equilibria of the game, one must study the sign of the profit differentials Δ O A = O i * I / N I − O i * N I / N I , Δ O B = O i * N I / I − O i * I / I and Δ O C = O i * N I / N I − O i * I / I for i={1, 2},i≠j. Under the conditions of stability and R&D cost, we have that ΔO_A > 0, ΔO_B < 0 and ΔO_C < 0. Therefore, (I,I) is the unique Pareto efficient Nash equilibrium, and the R&D investment decision game is an anti-prisoner’s dilemma.

References

Atallah, G. 2004. “The Protection of Innovations.” CIRANO Working Paper 2004s-02. Available at: https://cirano.qc.ca/files/publications/2004s-02.pdf.Search in Google Scholar

Azar, J., M. C. Schmalz, and I. Tecu. 2018. “Anticompetitive Effects of Common Ownership.” The Journal of Finance 73 (4): 1513–65. https://doi.org/10.1111/jofi.12698.Search in Google Scholar

Bárcena-Ruiz, J. C., and M. B. Garzón. 2020. “Disclosure of R&D Knowledge in a Mixed Duopoly.” The Manchester School 88 (4): 584–98.10.1111/manc.12309Search in Google Scholar

Bárcena-Ruiz, J. C., and A. Sagasta. 2021. “Environmental Policies with Consumer-Friendly Firms and Cross-Ownership.” Economic Modelling 103: 105612. https://doi.org/10.1016/j.econmod.2021.105612.Search in Google Scholar

Bettis, R. A., and M. A. Hitt. 1995. “The New Competitive Landscape.” Strategic Management Journal 16 (S1): 7–19. https://doi.org/10.1002/smj.4250160915.Search in Google Scholar

Brito, D., R. Ribeiro, and H. Vasconcelos. 2020. “Overlapping Ownership, Endogenous Quality, and Welfare.” Economics Letters 190: 109074. https://doi.org/10.1016/j.econlet.2020.109074.Search in Google Scholar

Buccella, D., L. Fanti, and L. Gori. 2023a. “The R&D Investment Decision Game with Product Differentiation.” The BE Journal of Theoretical Economics 23 (2): 601–37.10.1515/bejte-2021-0129Search in Google Scholar

Buccella, D., L. Fanti, and L. Gori. 2023b. “Network Externalities, Product Compatibility and Process Innovation.” Economics of Innovation and New Technology 32 (8): 1156–89. https://doi.org/10.1080/10438599.2022.2095513.Search in Google Scholar

Buccella, D., L. Fanti, and L. Gori. 2023c. “Optimal R&D Disclosure in Network Industries.” Economic Systems 47 (4): 101144.10.1016/j.ecosys.2023.101144Search in Google Scholar

Buccella, D., L. Fanti, and L. Gori. 2023d. “The Disclosure Decision Game: Subsidies and Incentives for R&D Activity.” Mathematical Social Sciences 125: 11–26.10.1016/j.mathsocsci.2023.06.002Search in Google Scholar

Chen, L., T. Matsumura, and C. Zeng. 2024. “Welfare Consequence of Common Ownership in a Vertically Related Market.” The Journal of Industrial Economics 72 (2): 996–1004. https://doi.org/10.1111/joie.12380.Search in Google Scholar

d’Aspremont, C., and A. Jacquemin. 1988. “Cooperative and Noncooperative R&D in Duopoly with Spillovers.” The American Economic Review 78 (5): 1133–7.Search in Google Scholar

de Haas, S., and J. Paha. 2016. “Partial Cross Ownership and Collusion.” MAGKS Joint Discussion Paper Series in Economics. No. 32-2016. Available at: https://www.econstor.eu/bitstream/10419/146639/1/866208380.pdf.Search in Google Scholar

Dong, Q., and J. C. Bárcena-Ruiz. 2021. “Corporate Social Responsibility and Disclosure of R&D Knowledge.” Economics of Innovation and New Technology 30 (6): 585–602.10.1080/10438599.2020.1741186Search in Google Scholar

Farrell, J., and C. Shapiro. 1990. “Asset Ownership and Market Structure in Oligopoly.” The RAND Journal of Economics 21 (2): 275–92, https://doi.org/10.2307/2555424.Search in Google Scholar

Gersbach, H., and A. Schmutzler. 2003. “Endogenous Spillovers and Incentives to Innovate.” Economic Theory 21 (1): 59–79, https://doi.org/10.1007/s00199-001-0245-8.Search in Google Scholar

Gil-Moltó, M. J., J. Poyago-Theotoky, and V. Zikos. 2011. “R&D Subsidies, Spillovers, and Privatization in Mixed Markets.” Southern Economic Journal 78 (1): 233–55.10.4284/0038-4038-78.1.233Search in Google Scholar

Gilo, D., Y. Moshe, and Y. Spiegel. 2006. “Partial Cross Ownership and Tacit Collusion.” The Rand Journal of Economics 37 (1): 81–99. https://doi.org/10.1111/j.1756-2171.2006.tb00005.x.Search in Google Scholar

Kang, J. K., J. Luo, and H. S. Na. 2018. “Are Institutional Investors with Multiple Blockholdings Effective Monitors?” Journal of Financial Economics 128 (3): 576–602. https://doi.org/10.1016/j.jfineco.2018.03.005.Search in Google Scholar

Lambertini, L., F. Lotti, and E. Santarelli. 2004. “Infra-industry Spillovers and R&D Cooperation: Theory and Evidence.” Economics of Innovation and New Technology 13 (4): 311–28.10.1080/10438590410001629016Search in Google Scholar

López, Á. L., and X. Vives. 2019. “Overlapping Ownership, R&D Spillovers, and Antitrust Policy.” Journal of Political Economy 127 (5): 2394–437.10.1086/701811Search in Google Scholar

Matsumura, T. 1998. “Partial Privatization in Mixed Duopoly.” Journal of Public Economics 70 (3): 473–83. https://doi.org/10.1016/s0047-2727-98-00051-6.Search in Google Scholar

Milliou, C. 2009. “Endogenous Protection of R&D Investments.” Canadian Journal of Economics/Revue canadienne d’économique 42 (1): 184–205.10.1111/j.1540-5982.2008.01504.xSearch in Google Scholar

Piga, C., and J. Poyago-Theotoky. 2005. “Endogenous R&D Spillovers and Locational Choice.” Regional Science and Urban Economics 35 (2): 127–39.10.1016/j.regsciurbeco.2004.02.002Search in Google Scholar

Poyago-Theotoky, J. 1999. “A Note on Endogenous Spillovers in a Non-tournament R&D Duopoly.” Review of Industrial Organization 15 (3): 253–62.10.1023/A:1007732218120Search in Google Scholar

Reynolds, R. J., and B. R. Snapp. 1986. “The Competitive Effects of Partial Equity Interests and Joint Ventures.” International Journal of Industrial Organization 4 (2): 141–53. https://doi.org/10.1016/0167-7187-86-90027-5.Search in Google Scholar

Shy, O., and R. Stenbacka. 2020. “Common Ownership, Institutional Investors, and Welfare.” Journal of Economics & Management Strategy 29 (3): 706–23. https://doi.org/10.1111/jems.12380.Search in Google Scholar

Teece, D. J. 1982. “Towards an Economic Theory of the Multiproduct Firm.” Journal of Economic Behavior & Organization 3 (1): 39–63. https://doi.org/10.1016/0167-2681-82-90003-8.Search in Google Scholar

Vives, X. 2020. “Common Ownership, Market Power, and Innovation.” International Journal of Industrial Organization 70: 102528. https://doi.org/10.1016/j.ijindorg.2019.102528.Search in Google Scholar

Xing, M., and S. H. Lee. 2024a. “Consumer Environmental Awareness in a Green Managerial Delegation Contract under Common Ownership.” Metroeconomica. Available at: https://onlinelibrary.wiley.com/doi/10.1111/meca.12492.10.1111/meca.12492Search in Google Scholar

Xing, M., and S. H. Lee. 2024b. “Cross-ownership and Strategic Environmental Corporate Social Responsibility under Price Competition.” Environment and Development Economics 29 (3): 234–56. https://doi.org/10.1017/s1355770x24000032.Search in Google Scholar

Xing, M., L. F. Wang, and C. Zhou. 2024. “Cross-ownership on R&D and Social Welfare in Mixed Oligopoly.” Economics of Innovation and New Technology 33 (2): 206–17.10.1080/10438599.2022.2163480Search in Google Scholar

Received: 2024-06-25

Accepted: 2025-06-04

Published Online: 2025-07-07

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https://doi.org/10.1515/bejeap-2024-0215

Keywords for this article

R&D knowledge disclosure; cross-ownership; mixed duopoly