Cross-ownership in duopoly: Are there any incentives to divest?

Therefore, the effect due to follower’s strategic response has two components: First, the effect due to internalization of the follower’s strategic response by the leader while choosing its strategy - the second term of equation (7) -, which is absent under simultaneous move competition; and second, the effect due to the follower’s strategic response even in absence of internalization by the leade. [...] The first component of the effect due to follower’s strategic response is negative (positive) in the case of strategic substitutes (strategic complements) and is opposite in sign to that of the second term. [...] 2 1 2 1 1 2 1 𝜕2Γ 𝜕2Γ 𝜕2Γ 𝜕2Γ Note that since (i) 𝑖 < 0 and | 𝑖2 2 | > | 𝑖 | > | 𝑖2 | by Assumption A2, and (ii) 𝜕𝑔𝑖 𝜕𝑔𝑖 𝜕𝑔𝑖𝜕𝑔𝑗 𝜕𝑔𝑗 𝜕𝑔 2 2 2 2 | 2,𝑅𝐹 𝜕 Γ 𝜕 Γ 𝜕𝑔 | < 1 ( 1 + 1 2,𝑅𝐹 𝜕 Γ 𝜕 Γ 𝜕𝑔2,𝑅𝐹 𝜕𝑔2,𝑅𝐹 (as shown above), we get 2 ) + ( 1 + 12 ) < 0. [...] If > 0 and < 0, then 1 > 0 if and only if 2 + 2 2,𝑅𝐹 < 0, or else, 𝜕𝑔1𝜕𝑔2 𝜕𝑔𝑗 𝜕𝑟1 𝜕𝑔1 𝜕𝑔2 𝜕𝑔1 𝜕𝜋2 𝜕𝑔2,𝑅𝐹 𝜕𝑔 𝜕𝑔< − 1 = 2 𝜕𝜋 ⌋ (since by (A3), 2 𝜕𝜋 𝜕𝜋 = − (𝑠 1 𝜕𝑔 2 𝜕𝑔 𝜕𝑔 2 − 𝑟2) > 0), with 1 1 𝜕𝑔 𝜋2=𝑐𝑜𝑛𝑠𝑡 2 𝜕𝑔2 2 𝜕𝑔2⌋ the slope of firm 2’s iso-profit curve in the 𝑔1𝑔2-plane. [...] If 2 > 0 and 𝑖 > 0, then 1 < 0 if and only if 2 + 2 > 0, or esle, 𝜕𝑔1𝜕𝑔2 𝜕𝑔𝑗 𝜕𝑟1 𝜕𝑔1 𝜕𝑔2 𝜕𝑔1 𝜕𝜋2 𝜕𝑔2,𝑅𝐹 𝜕𝑔 𝜕𝑔 𝜕𝜋 𝜕𝜋< − 1 2𝜕𝜋 = ⌋ (since by (A3), 2 = − (𝑠2 − 𝑟2) 1 < 0.) 𝜕𝑔1 2 𝜕𝑔1 𝜕𝑔2 𝜕𝑔2 𝜕𝑔 𝜋2=𝑐𝑜𝑛𝑠𝑡2 To be able to draw clear conclusions in cases 3 and 4, we first prove the following Lemma.