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

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.

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^′[Φ⁻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)

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