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How resource information backgrounds trigger post-merger integration and technology innovation? A dynamic analysis of resource similarity and complementarity

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Abstract

Overseas mergers and acquisitions (M&A) proposed by companies from emerging economies have been aiming to secure outward technology sourcing from developed countries in order to improve their technology innovation abilities in recent years. This paper proposes a comprehensive analytical framework of post-merger integration’s influence on technology innovation by global game modeling. We show how different resource similarity and resource complementarity backgrounds of the acquirer and target companies can affect post-merger strategies and technology innovation output through multi-stage analysis with an asymmetrical payoff structure. We focus on two main dimensions of post-merger integration, which are integration degree and target autonomy. Equilibrium analysis that is based on potential innovation output signals show that resource similarity has a positive relation with integration and a negative relation with target autonomy in overseas M&A; however, resource complementarity has the opposite effects compared with resource similarity. The positive interaction between resource similarity and complementarity will trigger more M&A and increase the degrees of integration and autonomy; M&A integration has a positive impact on technology innovation output. The innovation growth of the acquiring company is affected by the effectiveness of the post-merger process and the interaction of substitution elasticity with resource potential difference. Our study provides insight into the factors driving post-merger decisions and contributes to a multi-stage resource-based understanding of technology innovation induced by overseas post-merger integration.

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Notes

  1. The global game is first studied by Carlsson and van Damme (1993). It models economic environment with uncertain economic fundamentals summarized by a state theta and each player observes a different signal of state with a small amount of noise.

  2. Latin Hypercube Sampling is used to test the effects of jointly varying the parameters in the model (Scott et al. 2016).

  3. In their empirical analysis of knowledge driving M&As, the marginal contribution of technology similarity is 6.64 − 3.164*2*sim. The marginal contribution of technology complementary is 3.772; along with the descriptive statistics data provided in theirs, the mean value of technology similarity is 0.45. By calculation, technology similarity has a marginal contribution of 3.7924, which is very close to that of technology complementarity.

  4. The Boston Consulting Group (BCG): Gearing up new-era China’s outbound M&A Sep 2015 CHN.

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Acknowledgments

The authors are grateful for the support of key project of the National Social Science Fund (14AJY007), Key Projects of Zhejiang Province Natural Science (LZ14G020002), CRPE China private economy development research (2014).

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Correspondence to Feiqiong Chen.

Appendices

Appendices

1.1 Appendix 1: Proof of Proposition 1

Build the following function:

$${\text{h}}\left( {\tilde{\uptheta }_{\text{A}} } \right) = \tilde{\uptheta }_{\text{A}} - {\text{C}} - {\text{K}} - \uppi^{\text{sA}} {\text{R}}^{\text{A}} + \uppi^{\text{sA}} {\text{R}}^{\text{B}} - \uppi^{\text{sA}} {\text{R}}^{\text{B}} \Phi \left\{ {\upalpha \sqrt {\frac{1}{\upbeta }} \left( {\tilde{\uptheta }_{\text{A}} - {\text{y}}} \right) - \sqrt {1 + \frac{\upalpha }{\upbeta }} \Phi^{ - 1} \left[ {1 - \frac{\text{V}}{{{\text{d}}\left( {{\text{V}} + \upkappa \upvarphi - {\text{d}}\uptau } \right)}}} \right]} \right\}$$

Differentiate with respect to \(\tilde{\uptheta }_{\text{A}}\):

$$\frac{{\partial {\text{h}}\left( {\tilde{\uptheta }_{\text{A}} } \right)}}{{\partial \tilde{\uptheta }_{\text{A}} }} = 1 - \uppi^{\text{sA}} {\text{R}}^{\text{B}} \frac{\upalpha }{\sqrt \upbeta }\emptyset ({\text{z}}1)$$
(29)

where \({\text{z}}1 = \upalpha \sqrt {\frac{1}{\upbeta }} \left( {\tilde{\uptheta }_{\text{A}} - {\text{y}}} \right) - \sqrt {1\,+\, \frac{\upalpha }{\upbeta }} \Phi^{ - 1} \left[ {1 - \frac{\text{V}}{{{\text{d}}\left( {{\text{V}}\,+\,\upkappa \upvarphi \,-\, {\text{d}}\uptau } \right)}}} \right], \, \emptyset\) is pdf of normal distribution. \(\emptyset \left( {{\text{z}}1} \right) \le \frac{1}{{\sqrt {2\uppi } }}\), when \(\uppi^{\text{sA}} {\text{R}}^{\text{B}} < \frac{{\sqrt {2\upbeta \uppi } }}{\upalpha }\), we have \(\frac{{\partial {\text{h}}\left( {\tilde{\uptheta }_{\text{A}} } \right)}}{{\partial \tilde{\uptheta }_{\text{A}} }} > 0\), which means unique solution exists. Similarly,

$$\begin{aligned} {\text{g}}\left( {\tilde{\uptheta }_{\text{B}} } \right) & = \tilde{\uptheta }_{\text{B}} + {\text{K}} - \uppi^{\text{sB}} {\text{R}}^{\text{B}} \Phi \left\{ {\upalpha \sqrt {\frac{1}{\upbeta }} \left( {\tilde{\uptheta }_{\text{B}} - {\text{y}}} \right) - \sqrt {1 + \frac{\upalpha }{\upbeta }} \Phi^{ - 1} \left[ {1 - \frac{\text{V}}{{{\text{d}}\left( {{\text{V}} - \upkappa \updelta - {\text{d}}\uptau + {\text{M}}\uptau } \right)}}} \right]} \right\} \\ {\text{z}}11 & = \upalpha \sqrt {\frac{1}{\upbeta }} \left( {\tilde{\uptheta }_{\text{B}} - {\text{y}}} \right) - \sqrt {1 + \frac{\upalpha }{\upbeta }} \Phi^{ - 1} \left[ {1 - \frac{\text{V}}{{{\text{d}}\left( {{\text{V}} - \upkappa \updelta - {\text{d}}\uptau + {\text{M}}\uptau } \right)}}} \right] \\ \frac{{\partial {\text{g}}\left( {\tilde{\uptheta }_{\text{B}} } \right)}}{{\partial \tilde{\uptheta }_{\text{B}} }} & = 1 - \uppi^{\text{sB}} {\text{R}}^{\text{B}} \frac{\upalpha }{\sqrt \upbeta }\emptyset ({\text{z}}11) \\ \end{aligned}$$
(30)

When \(\uppi^{\text{sB}} {\text{R}}^{\text{B}} < \frac{{\sqrt {2\upbeta \uppi } }}{\upalpha }\), we have \(\frac{{\partial {\text{g}}\left( {\tilde{\uptheta }_{\text{B}} } \right)}}{{\partial \tilde{\uptheta }_{\text{B}} }} > 0\). To sum up, when \(\uppi^{\text{sA}} {\text{R}}^{\text{B}} < \frac{{\sqrt {2\upbeta \uppi } }}{\upalpha }\), both companies have unique equilibrium.

$$\frac{{\partial \tilde{\uptheta }_{\text{A}} }}{\partial \upkappa } = - \frac{{\uppi^{\text{sA}} {\text{R}}^{\text{B}} \emptyset \left( {{\text{z}}1} \right)\sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi^{ - 1} \left( {{\text{z}}2} \right)} \right]}}\frac{{{\text{Vd}}\upvarphi }}{{\left[ {{\text{d}}\left( {{\text{V}} + \upkappa \upvarphi - {\text{d}}\uptau } \right)} \right]^{2} }}}}{{1 - \uppi^{\text{sA}} {\text{R}}^{\text{B}} \frac{\upalpha }{\sqrt \upbeta }\emptyset \left( {{\text{z}}1} \right)}}$$
(31)

where \({\text{z}}2 = 1 - \frac{\text{V}}{{{\text{d}}({\text{V}} + \upkappa \upvarphi - {\text{d}}\uptau )}}\), \(\frac{{\partial \tilde{\uptheta }_{\text{A}} }}{\partial \upkappa } < 0\)

$$\frac{{\partial \tilde{\uptheta }_{\text{A}} }}{{\partial {\text{d}}}} = - \frac{{\uppi^{\text{sA}} {\text{R}}^{\text{B}} \emptyset \left( {{\text{z}}1} \right)\sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi^{ - 1} \left( {{\text{z}}2} \right)} \right]}}\frac{{{\text{V}}({\text{V}} + \upkappa \upvarphi - 2{\text{d}}\uptau )}}{{\left[ {{\text{d}}\left( {{\text{V}} + \upkappa \upvarphi - {\text{d}}\uptau } \right)} \right]^{2} }}}}{{1 - \uppi^{\text{sA}} {\text{R}}^{\text{B}} \frac{\upalpha }{\sqrt \upbeta }\emptyset \left( {{\text{z}}1} \right)}}$$
(32)

V + κφ − 2dτ < 0 when \(\uppi^{\text{sA}} {\text{R}}^{\text{B}} < \frac{{\sqrt {2\upbeta \uppi } }}{\upalpha }\), \(\frac{{\partial \tilde{\uptheta }_{\text{A}} }}{{\partial {\text{d}}}} > 0\).

$$\tilde{\Theta }_{\text{B}} = - {\rm K} + \uppi^{\text{sB}} {\text{R}}^{\text{B}} \Phi \left\{ {\upalpha \sqrt {\frac{1}{\upbeta }} \left( {\tilde{\uptheta }_{\text{B}} - {\text{y}}} \right) - \sqrt {1 + \frac{\upalpha }{\upbeta }} \Phi^{ - 1} \left[ {1 - \frac{\text{V}}{{{\text{d}}\left( {{\text{V}} - \upkappa \updelta - {\text{d}}\uptau + {\text{M}}\uptau } \right)}}} \right]} \right\}$$
$$\frac{{\partial \tilde{\uptheta }_{\text{B}} }}{\partial \upkappa } = - \frac{{\uppi^{\text{sB}} {\text{R}}^{\text{B}} \emptyset \left( {{\text{z}}11} \right)\sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi^{ - 1} \left( {{\text{z}}22} \right)} \right]}}\frac{{ - {\text{Vd}}\updelta }}{{\left[ {{\text{d}}\left( {{\text{V}} - \upkappa \updelta - {\text{d}}\uptau + {\text{M}}\uptau } \right)} \right]^{2} }}}}{{1 - \uppi^{\text{sB}} {\text{R}}^{\text{B}} \frac{\upalpha }{\sqrt \upbeta }\emptyset \left( {{\text{z}}11} \right)}}$$
(33)
$${\text{z}}11 = \upalpha \sqrt {\frac{1}{\upbeta }} \left( {\tilde{\uptheta }_{\text{B}} - {\text{y}}} \right) - \sqrt {1 + \frac{\upalpha }{\upbeta }} \Phi^{ - 1} \left[ {1 - \frac{\text{V}}{{{\text{d}}\left( {{\text{V}} - \upkappa \updelta - {\text{d}}\uptau + {\text{M}}\uptau } \right)}}} \right],\quad {\text{z}}22 = 1 - \frac{\text{V}}{{{\text{d}}({\text{V}} - \upkappa \updelta - {\text{d}}\uptau + {\text{M}}\uptau )}}$$
$$\frac{{\partial \tilde{\uptheta }_{\text{B}} }}{{\partial {\text{d}}}} = - \frac{{\uppi^{\text{sB}} {\text{R}}^{\text{B}} \emptyset \left( {{\text{z}}11} \right)\sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi^{ - 1} \left( {{\text{z}}22} \right)} \right]}}\frac{{{\text{V}}[{\text{V}} - \upkappa \updelta + 2\left( {{\text{M}} - {\text{d}}} \right)\uptau ]}}{{\left[ {{\text{d}}\left( {{\text{V}} - \upkappa \updelta - {\text{d}}\uptau + {\text{M}}\uptau } \right)} \right]^{2} }}}}{{1 - \uppi^{\text{sB}} {\text{R}}^{\text{B}} \frac{\upalpha }{\sqrt \upbeta }\emptyset \left( {{\text{z}}11} \right)}}$$
(34)

Thus \(\frac{{\partial \tilde{\uptheta }_{\text{B}} }}{\partial \upkappa } > 0.{\text{V}} - \upkappa \updelta - {\text{d}}\uptau + {\text{M}}\uptau > \frac{\text{V}}{\text{d}} > 0\), M > d, V − κδ + 2(M − d)τ > 0. \(\frac{{\partial \tilde{\uptheta }_{\text{B}} }}{{\partial {\text{d}}}} < 0\).

1.2 Appendix 2: Proof of Proposition 2

$$\frac{{\partial \tilde{\uptheta }_{\text{A}} }}{\partial \upkappa } = - \frac{{\uppi^{\text{sA}} {\text{R}}^{\text{B}} \emptyset \left( {{\text{z}}1} \right)\sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi^{ - 1} \left( {{\text{z}}2} \right)} \right]}}\frac{{{\text{Vd}}\upvarphi }}{{\left[ {{\text{d}}\left( {{\text{V}} + \upkappa \upvarphi - {\text{d}}\uptau } \right)} \right]^{2} }}}}{{1 - \uppi^{\text{sA}} {\text{R}}^{\text{B}} \frac{\upalpha }{\sqrt \upbeta }\emptyset \left( {{\text{z}}1} \right)}}$$

where \(\frac{{\partial {\text{z}}2}}{\partial \upkappa } = \frac{{{\text{V}}\upvarphi }}{{{\text{d}}({\text{V}} + \upkappa \upvarphi - {\text{d}}\uptau )^{2} }} > 0\), \(\frac{{\partial {\text{z}}2}}{{\partial {\text{d}}}} = \frac{{{\text{V}}({\text{V}} + \upkappa \upvarphi - 2{\text{d}}\uptau )}}{{[{\text{d}}({\text{V}} + \upkappa \upvarphi - {\text{d}}\uptau )]^{2} }} < 0\), \(\frac{{\partial {\text{z}}1}}{{\partial {\text{d}}}} = \upalpha \sqrt {\frac{1}{\upbeta }} \frac{{\partial \tilde{\uptheta }_{\text{A}} }}{{\partial {\text{d}}}} - \sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi^{ - 1} \left( {{\text{z}}2} \right)} \right]}}\frac{{\partial {\text{z}}2}}{{\partial {\text{d}}}} > 0\),

$$\begin{aligned} \frac{{\partial \tilde{\uptheta }_{{\text{A}}} }}{{\partial \upkappa \partial {\text{d}}}} &= - \left\{ {\left\{ {\uppi ^{{{\text{sA}}}} {\text{R}}^{{\text{B}}} \emptyset ^{'} \left( {{\text{z}}1} \right)\frac{{\partial {\text{z}}1}}{{\partial {\text{d}}}}\sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}2} \right)} \right]}}\frac{{\partial {\text{z}}2}}{{\partial \upkappa }}} \right.} \right. \hfill \\ & \quad + \uppi ^{{{\text{sA}}}} {\text{R}}^{{\text{B}}} \emptyset \left( {{\text{z}}1} \right)\sqrt {1 + \frac{\upalpha }{\upbeta }} \left[ { - \frac{1}{{\left\{ {\emptyset \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}2} \right)} \right]} \right\}^{2} }}} \right]\emptyset ^{'} \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}2} \right)} \right]\frac{1}{{\emptyset \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}2} \right)} \right]}}\frac{{\partial {\text{z}}2}}{{\partial {\text{d}}}}\frac{{\partial {\text{z}}2}}{{\partial \upkappa }} \hfill \\ & \left. {\quad +\, \uppi ^{{{\text{sA}}}} {\text{R}}^{{\text{B}}} \emptyset \left( {{\text{z}}1} \right)\sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}2} \right)} \right]}}\frac{{\partial \left( {\frac{{\partial {\text{z}}2}}{{\partial \upkappa }}} \right)}}{{\partial {\text{d}}}}} \right\}{\text{v}} \hfill \\ & {{\left. {\quad - u\left( { - \uppi ^{{{\text{sA}}}} {\text{R}}^{{\text{B}}} \emptyset ^{'} \left( {{\text{z}}1} \right)\frac{{\partial {\text{z}}1}}{{\partial {\text{d}}}}\sqrt {1 + \frac{\upalpha }{\upbeta }} } \right)} \right\}} \mathord{\left/ {\vphantom {{\left. {\quad - u\left( { - \uppi ^{{{\text{sA}}}} {\text{R}}^{{\text{B}}} \emptyset ^{'} \left( {{\text{z}}1} \right)\frac{{\partial {\text{z}}1}}{{\partial {\text{d}}}}\sqrt {1 + \frac{\upalpha }{\upbeta }} } \right)} \right\}} {{\text{v}}^{2} }}} \right. \kern-\nulldelimiterspace} {{\text{v}}^{2} }} \hfill \\ \end{aligned}$$
(35)

When y is larger enough, z1 < 0, \(\emptyset^{'} \left( {z1} \right) > 0\). \({\text{z}}2 \in (0,0.5)\), Φ−1(z2) < 0, θ−1(z2)] > 0.

$${\frac{{\partial \left( {\frac{{\partial {\text{z}}2}}{\partial \upkappa }} \right)}}{{\partial {\text{d}}}} = - \frac{{{\text{V}}\upvarphi [({\text{V}} + \upkappa \upvarphi - {\text{d}}\uptau )^{2} - 2{\text{d}}\uptau \left( {{\text{V}} + \upkappa \upvarphi - {\text{d}}\uptau } \right)]}}{{[{\text{d}}({\text{V}} + \upkappa \upvarphi - {\text{d}}\uptau )]^{2} }} = - \frac{{{\text{V}}\upvarphi [\left( {{\text{V}} + \upkappa \upvarphi - 3{\text{d}}\uptau } \right)]}}{{{\text{d}}^{2} ({\text{V}} + \upkappa \upvarphi - {\text{d}}\uptau )^{3} }} > 0,\; \frac{{\partial \tilde{\uptheta }_{\text{A}} }}{{\partial \upkappa \partial {\text{d}}}} < 0.}$$
$$\frac{{\partial {\text{z}}22}}{\partial \upkappa } = \frac{{ - {\text{V}}\updelta }}{{{\text{d}}({\text{V}} - \upkappa \updelta - {\text{d}}\uptau + {\text{M}}\uptau )^{2} }} < 0,\;\frac{{\partial {\text{z}}22}}{{\partial {\text{d}}}} > 0,\;\frac{{\partial {\text{z}}11}}{{\partial {\text{d}}}} = \upalpha \sqrt {\frac{1}{\upbeta }} \frac{{\partial \tilde{\uptheta }_{\text{B}} }}{{\partial {\text{d}}}} - \sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi^{ - 1} \left( {{\text{z}}22} \right)} \right]}}\frac{{\partial {\text{z}}22}}{{\partial {\text{d}}}} < 0.$$
$$\begin{aligned} \frac{{\partial \tilde{\uptheta }_{{\text{B}}} }}{{\partial \upkappa \partial {\text{d}}}} & = - \left\{ {\left\{ {\uppi ^{{{\text{sA}}}} {\text{R}}^{{\text{B}}} \emptyset ^{'} \left( {{\text{z}}11} \right)\frac{{\partial {\text{z}}11}}{{\partial {\text{d}}}}\sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}22} \right)} \right]}}\frac{{\partial {\text{z}}22}}{{\partial \upkappa }}~} \right.} \right. \hfill \\ & \quad + \uppi ^{{{\text{sA}}}} {\text{R}}^{{\text{B}}} \emptyset \left( {{\text{z}}11} \right)\sqrt {1 + \frac{\upalpha }{\upbeta }} \left[ { - \frac{1}{{\left\{ {\emptyset \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}22} \right)} \right]} \right\}^{2} }}} \right] \times \emptyset ^{'} \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}22} \right)} \right]\frac{1}{{\emptyset \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}22} \right)} \right]}}\frac{{\partial {\text{z}}22}}{{\partial {\text{d}}}}\frac{{\partial {\text{z}}22}}{{\partial \upkappa }} \hfill \\ & \quad \left. { +\, \uppi ^{{{\text{sA}}}} {\text{R}}^{{\text{B}}} \emptyset \left( {{\text{z}}11} \right)\sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}22} \right)} \right]}}\frac{{\partial \left( {\frac{{\partial {\text{z}}22}}{{\partial \upkappa }}} \right)}}{{\partial {\text{d}}}}} \right\}{\text{v}} \hfill \\ & {{\left. {\quad -\, {\text{u}}\left( { - \uppi ^{{{\text{sA}}}} {\text{R}}^{{\text{B}}} \emptyset ^{'} \left( {{\text{z}}11} \right)\frac{{\partial {\text{z}}11}}{{\partial {\text{d}}}}\sqrt {1 + \frac{\upalpha }{\upbeta }} } \right)} \right\}} \mathord{\left/ {\vphantom {{\left. {\quad - {\text{u}}\left( { - \uppi ^{{{\text{sA}}}} {\text{R}}^{{\text{B}}} \emptyset ^{'} \left( {{\text{z}}11} \right)\frac{{\partial {\text{z}}11}}{{\partial {\text{d}}}}\sqrt {1 + \frac{\upalpha }{\upbeta }} } \right)} \right\}} {{\text{v}}^{2} }}} \right. \kern-\nulldelimiterspace} {{\text{v}}^{2}}}\end{aligned}$$
(36)

z11 < 0, θ′(z11) > 0. z22 ∈ (0, 0.5), Φ−1(z22) < 0, θ′[Φ−1(z22)] > 0. \(\frac{{\partial \left( {\frac{{\partial {\text{z}}22}}{\partial \upkappa }} \right)}}{{\partial {\text{d}}}} = - \frac{{{\text{V}}\updelta [\left( {{\text{V}} - \upkappa \upvarphi - 3{\text{d}}\uptau + 3{\text{M}}\uptau } \right)]}}{{{\text{d}}^{2} ({\text{V}} - \upkappa \updelta - {\text{d}}\uptau + {\text{M}}\uptau )^{3} }} > 0. \quad \frac{{\partial \tilde{\uptheta }_{\text{B}} }}{{\partial \upkappa \partial {\text{d}}}} < 0.\)

1.3 Appendix 3: Proof of Lemma 1

$$\frac{{\partial \uplambda \left( {\uptheta_{\text{A}} } \right)}}{\partial k} = - \emptyset \left( {{\text{z}}1} \right)\left\{ {\frac{\upalpha + \upbeta }{\sqrt \upbeta }\frac{{\partial \tilde{\uptheta }_{\text{A}} }}{\partial \upkappa } - \sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi^{ - 1} \left( {{\text{z}}2} \right)} \right]}}\frac{{{\text{Vd}}\upvarphi }}{{\left[ {{\text{d}}\left( {{\text{V}} + \upkappa \upvarphi - {\text{d}}\uptau } \right)} \right]^{2} }}} \right\}$$
(37)

The first one in the braces is negative. The second one is positive, and \(\frac{{\partial \uplambda \left( {\uptheta_{\text{A}} } \right)}}{{\partial {\text{k}}}} > 0\).

$$\frac{{\partial \uplambda \left( {\uptheta_{\text{A}} } \right)}}{{\partial {\text{d}}}} = - \emptyset \left( {{\text{z}}1} \right)\left\{ {\frac{\upalpha + \upbeta }{\sqrt \upbeta }\frac{{\partial \tilde{\uptheta }_{\text{A}} }}{{\partial {\text{d}}}} - \sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi^{ - 1} \left( {{\text{z}}2} \right)} \right]}}\frac{{{\text{V}}({\text{V}} + \upkappa \upvarphi - 2{\text{d}}\uptau )}}{{\left[ {{\text{d}}\left( {{\text{V}} + \upkappa \upvarphi - {\text{d}}\uptau } \right)} \right]^{2} }}} \right\}$$
(38)

The first one in the braces is positive. The second one is negative, and \(\frac{{\partial \uplambda \left( {\uptheta_{\text{A}} } \right)}}{{\partial {\text{d}}}} < 0\).

1.4 Appendix 4: Proof of Proposition 3

$$\frac{{\partial {\text{I}}}}{\partial \uptheta } = \frac{{\partial \left( {\mathop \smallint \nolimits_{0}^{{\uptheta_{\text{A}} }} \uplambda \left( {\uptheta_{\text{A}} } \right){\text{d}}\uptheta_{\text{A}} } \right)}}{{\partial \uptheta_{\text{A}} }} = \uplambda \left( {\uptheta_{\text{A}} } \right)$$
(39)
$$\frac{{\partial {\text{I}}}}{\partial \upkappa } = \frac{{\partial \left( {\frac{{\partial {\text{I}}}}{\partial \uptheta }} \right)}}{\partial \upkappa } = \frac{{\partial \uplambda \left( {\uptheta_{\text{A}} } \right)}}{{\partial {\text{k}}}} > 0,\;\frac{{\partial {\text{I}}}}{{\partial {\text{d}}}} = \frac{{\partial \left( {\frac{{\partial {\text{I}}}}{\partial \uptheta }} \right)}}{{\partial {\text{d}}}} = \frac{{\partial \uplambda \left( {\uptheta_{\text{A}} } \right)}}{{\partial {\text{d}}}} < 0.$$

1.5 Appendix 5: Proof of Proposition 4

$$\frac{{\partial {\text{w}}\left( {\uptheta_{\text{B}} } \right)}}{{\partial {\text{k}}}} = - \emptyset \left( {{\text{z}}11} \right)\left\{ {\frac{\upalpha + \upbeta }{\sqrt \upbeta }\frac{{\partial \tilde{\uptheta }_{\text{B}} }}{\partial \upkappa } - \sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi^{ - 1} \left( {{\text{z}}22} \right)} \right]}}\frac{{ - V{\text{d}}\updelta }}{{\left[ {{\text{d}}\left( {{\text{V}} - \upkappa \updelta - {\text{d}}\uptau + {\text{M}}\uptau } \right)} \right]^{2} }}} \right\}$$
(40)

\(\frac{{\partial {\text{w}}\left( {\uptheta_{\text{B}} } \right)}}{{\partial {\text{k}}}} < 0\). Because \(\frac{{\partial \tilde{\uptheta }_{\text{B}} }}{\partial \upkappa } > 0\)

$$\frac{{\partial {\text{w}}\left( {\uptheta_{\text{B}} } \right)}}{{\partial {\text{d}}}} = - \emptyset \left( {{\text{z}}11} \right)\left\{ {\frac{\upalpha + \upbeta }{\sqrt \upbeta }\frac{{\partial \tilde{\uptheta }_{\text{B}} }}{{\partial {\text{d}}}} - \sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi^{ - 1} \left( {{\text{z}}22} \right)} \right]}}\frac{{{\text{V}}({\text{V}} - \upkappa \updelta - 2{\text{d}}\uptau + 2{\text{M}}\uptau )}}{{\left[ {{\text{d}}\left( {{\text{V}} - \upkappa \updelta - {\text{d}}\uptau + {\text{M}}\uptau } \right)} \right]^{2} }}} \right\}$$
(41)

V − κδ + 2(M – d)τ > 0, \(\frac{{\partial \tilde{\uptheta }_{\text{B}} }}{{\partial {\text{d}}}} < 0\), so \(\frac{{\partial {\text{w}}\left( {\uptheta_{\text{B}} } \right)}}{{\partial {\text{d}}}} > 0\).

1.6 Appendix 6: Proof of Proposition 5

$$\begin{aligned} \frac{{\partial \uplambda \left( {\uptheta _{{\text{A}}} } \right)}}{{\partial {\text{k}}\partial {\text{d}}}} &= - \emptyset ^{'} \left( {{\text{z}}1} \right)\frac{{\partial {\text{z}}1}}{{\partial {\text{d}}}}\left\{ {\frac{{\upalpha + \upbeta }}{{\sqrt \upbeta }}\frac{{\partial \tilde{\uptheta }_{{\text{A}}} }}{{\partial \upkappa }} - \sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}2} \right)} \right]}}\frac{{\partial {\text{z}}2}}{{\partial \upkappa }}} \right\} \hfill \\ & \quad - \emptyset \left( {z1} \right)\left\{ {\frac{{\upalpha + \upbeta }}{{\sqrt \upbeta }}\frac{{\partial \tilde{\uptheta }_{{\text{A}}} }}{{\partial \upkappa \partial {\text{d}}}}} \right. \hfill \\ & \quad - \sqrt {1 + \frac{\upalpha }{\upbeta }} \left[ { - \frac{1}{{\left\{ {\emptyset \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}2} \right)} \right]} \right\}^{2} }}} \right]\emptyset ^{'} \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}2} \right)} \right]\frac{1}{{\emptyset \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}2} \right)} \right]}}\frac{{\partial {\text{z}}2}}{{\partial {\text{d}}}}\frac{{\partial {\text{z}}2}}{{\partial \upkappa }} \hfill \\ & \left. {\quad - \sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}2} \right)} \right]}}\frac{{\partial \left( {\frac{{\partial {\text{z}}2}}{{\partial \upkappa }}} \right)}}{{\partial {\text{d}}}}} \right\} \end{aligned}$$
(42)

\(\frac{{\partial \uplambda \left( {\uptheta_{\text{A}} } \right)}}{{\partial {\text{k}}\partial {\text{d}}}} > 0\),

$$\begin{aligned} \frac{{\partial {\text{w}}\left( {\uptheta _{{\text{B}}} } \right)}}{{\partial {\text{k}}\partial {\text{d}}}} &= - \emptyset ^{\prime } \left( {{\text{z}}11} \right)\frac{{\partial {\text{z}}11}}{{\partial {\text{d}}}}\left\{ {\frac{{\upalpha + \upbeta }}{{\sqrt \upbeta }}\frac{{\partial \tilde{\uptheta }_{{\text{B}}} }}{{\partial \upkappa }} - \sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}22} \right)} \right]}}\frac{{\partial {\text{z}}22}}{{\partial \upkappa }}} \right\} \hfill \\ & \quad - \emptyset \left( {{\text{z}}11} \right)\left\{ {\frac{{\upalpha + \upbeta }}{{\sqrt \upbeta }}\frac{{\partial \tilde{\uptheta }_{{\text{B}}} }}{{\partial \upkappa \partial d}}} \right. \hfill \\ & \quad - \sqrt {1 + \frac{\upalpha }{\upbeta }} \left[ { - \frac{1}{{\left\{ {\emptyset \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}22} \right)} \right]} \right\}^{2} }}} \right]\emptyset ^{\prime } \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}22} \right)} \right]\frac{1}{{\emptyset \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}22} \right)} \right]}}\frac{{\partial {\text{z}}22}}{{\partial {\text{d}}}}\frac{{\partial {\text{z}}22}}{{\partial \upkappa }} \hfill \\ & \left. {\quad - \sqrt {1 + \frac{\upalpha }{\upbeta }} \frac{1}{{\emptyset \left[ {\Phi ^{{ - 1}} \left( {{\text{z}}22} \right)} \right]}}\frac{{\partial \left( {\frac{{\partial {\text{z}}22}}{{\partial \upkappa }}} \right)}}{{\partial {\text{d}}}}} \right\} \end{aligned}$$
(43)

\(\frac{{\partial {\text{w}}\left( {\uptheta_{\text{B}} } \right)}}{{\partial {\text{k}}\partial {\text{d}}}} > 0\).

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Chen, F., Meng, Q. & Li, F. How resource information backgrounds trigger post-merger integration and technology innovation? A dynamic analysis of resource similarity and complementarity. Comput Math Organ Theory 23, 167–198 (2017). https://doi.org/10.1007/s10588-016-9222-4

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