Inducing information transparency: The roles of gray market and dual-channel

Abstract

Information asymmetry is particularly common in the supply chain framework. As we know, the downstream retailer usually knows more about the market demand information than the upstream manufacturer because of her proximity to consumers. This paper considers a supply chain that a manufacturer sells products to consumers via a retailer in a local market. Besides, the manufacturer has an option to establish a direct online channel in the overseas market, in which the retailer can sell products as a gray marketer to earn an extra profit. Both markets’ demand information is transparent to the retailer but asymmetric to the manufacturer. There are two side an opposite effects of information sharing on the retailer. The positive effect is that when she withholds demand information she can possess an information advantage. However, the negative effect is that when the retailer shares her private demand information, especially when the manufacturer enters the overseas market, the retailer’s profit may be negatively affected because of the competition. Therefore, whether the retailer has motivation to share her private demand information with the manufacturer is an intriguing yet unanswered question. We first characterize the tradeoff in three supply chain structures, Local Market, Dual Market and Gray Market. Then, we obtain the equilibrium results of each case and find that a co-opetition strategy may arise because of competition when considering gray market and dual-channel. Finally, we uncover the underlying reasons and develop valuable insights.

Appendix

Proof of Lemma 5

In the case that the retailer does not share demand information, we derive the manufacturer’s payoff difference between entry and no entry by calculating Eqs. (19) and (21) as follows: $\pi _{M}^{NE}-\pi _{M}^{NN}=\frac{57{{a}^{2}}+38(\rho +2)ad+(9{{\rho }^{2}}+20\rho +28){{d}^{2}}}{152}-C-\frac{{{(2a+d+\rho d)}^{2}}}{16} =\frac{38{{a}^{2}}+76ad+(-{{\rho }^{2}}+2\rho +37){{d}^{2}}}{304}-C$. We define $\gamma (d)=\frac{38{{a}^{2}}+76ad+(-{{\rho }^{2}}+2\rho +37){{d}^{2}}}{304}$. Because $\frac{\partial \gamma (d)}{\partial d}=\frac{38a+(-{{\rho }^{2}}+2\rho +37)d}{152}>0$, the manufacturer will enter only when the entry cost C lower than $\gamma (d)$. $\square$

Proof of Lemma 6

In the case that the retailer shares demand information, we derive the manufacturer’s payoff differences between entry and no entry by calculating Eqs. (24) and (27) as follows: $\pi _{M}^{SE}-\pi _{M}^{SN}=\frac{57{{a}^{2}}+38(\rho +2)a\delta +(9{{\rho }^{2}}+20\rho +28){{\delta }^{2}}}{152}-C-\frac{{{(2a+3\delta -\rho \delta )}^{2}}+{{(2a+3\rho \delta -\delta )}^{2}}}{64} =\gamma (\delta )-C$. Noting that $\delta \in [0,\ 2d]$ and $\gamma (\delta )$ is increasing in $\delta$, we have when $C\le \gamma (0)$, the manufacturer will always enter; when $C>\gamma (2d)$, the manufacturer will never enter; when C is between $\gamma (0)$ and $\gamma (2d)$, the manufacturer will enter only if $\delta >{\hat{\delta }}(C)$. $\square$

Proof of Lemma 7

In order to examine the retailer’s information sharing strategy, we derive the retailer’s payoff differences between information sharing and non-information sharing by calculating Eqs. (29) and (30) as follows: When $C\le \gamma (0)$, $\pi _{R}^{S}(C)-\pi _{R}^{N}(C)=\frac{{{(19a+29\rho d-10d)}^{2}}+144{{d}^{2}}{{(1-\rho )}^{2}}}{5776}+\frac{144{{(1-\rho )}^{2}}+{{(29\rho -10)}^{2}}}{17328}{{d}^{2}}-\frac{{{(19a+29\rho d-10d)}^{2}}+144{{d}^{2}}{{(1-\rho )}^{2}}}{5776}-\frac{(1+{{\rho }^{2}}){{d}^{2}}}{12}=\frac{-459{{\rho }^{2}}-3756\rho -1200}{17328}{{d}^{2}}<0$. When $C>\gamma (2d)$, $\pi _{R}^{S}(C)-\pi _{R}^{N}(C)= \frac{{{(2a+3d-\rho d)}^{2}}+{{(2a+3\rho d-d)}^{2}}}{64}+\frac{5-6\rho +5{{\rho }^{2}}}{96}{{d}^{2}}-\frac{{{(2a+3d-\rho d)}^{2}}+{{(2a+3\rho d-d)}^{2}}}{64}-\frac{(1+{{\rho }^{2}}){{d}^{2}}}{12}=-\frac{{{(1+\rho )}^{2}}{{d}^{2}}}{32}<0$. Thus, the retailer will not share demand information with the manufacturer when the entry cost C is very low or especially high. $\square$

Proof of Proposition 1

When the retailer withholds her private demand information, the entry strategy of the manufacturer is that he will enter only when the entry cost C is lower than $\gamma (d)$. We derive the retailer’s profit difference between entry and no entry by calculating Eqs. (29) and (30) as follows: $\pi _{R}^{N}(C){{|}_{C\le \gamma (d)}}-\pi _{R}^{N}(C){{|}_{C>\gamma (d)}}=\frac{{{(19a+29\rho d-10d)}^{2}}+144{{d}^{2}}{{(1-\rho )}^{2}}}{5776}+\frac{(1+{{\rho }^{2}}){{d}^{2}}}{12} -\frac{{{(2a+3d-\rho d)}^{2}}+{{(2a+3\rho d-d)}^{2}}}{64}-\frac{(1+{{\rho }^{2}}){{d}^{2}}}{12}= \frac{-722{{a}^{2}}+(760\rho -2204)ad+(165{{\rho }^{2}}+430\rho -1317){{d}^{2}}}{11552}<0$. When the retailer shares her private demand information and the entry cost C is between $\gamma (0)$ and $\gamma (2d)$, then the manufacturer will enter only when the potential market demand is higher than the threshold ${\hat{\delta }}$. $\frac{\partial \pi _{R}^{S}}{\partial {\hat{\delta }}}=\frac{722{{a}^{2}}+(2204-760\rho )a{\hat{\delta }}+(1317-430\rho -165{{\rho }^{2}}){{{{\hat{\delta }}}}^{2}}}{11552}>0$. Recall that ${\hat{\delta }}(C)=\frac{\sqrt{38[{{a}^{2}}{{(1-\rho )}^{2}}-8C({{\rho }^{2}}-2\rho -37)]}-38a}{-{{\rho }^{2}}+2\rho +37}$ which is increasing in entry cost C. Hence, $\pi _{R}^{N}(C)$ and $\pi _{R}^{S}(C)$ are both increasing in C. Notably, there is a sudden upward increase for the retailer’s expected payoff in the non-sharing condition when $C=\gamma (d)$. To derive the retailer’s optimal information sharing strategy when $C\in (\gamma (0),\ \gamma (d)]$, it is necessary to compare the retailer’s payoff difference under sharing and non-sharing conditions when $C=\gamma (d)$.

Note that when $C=\gamma (d)$, ${\hat{\delta }}(C)=d$, $\phi (\rho )=\pi _{R}^{S}(\gamma (d))-\pi _{R}^{N}(\gamma (d))= \frac{2166{{a}^{2}}+(-1140\rho +3306)ad+(-2001{{\rho }^{2}}-3938\rho -3483){{d}^{2}}}{69312}$, $\phi '(\rho )=\frac{2166{{a}^{2}}+(-1140\rho +3306)ad+(-2001{{\rho }^{2}}-3938\rho -3483){{d}^{2}}}{69312} =\frac{-1140ad-4002\rho {{d}^{2}}-3938{{d}^{2}}}{69312}<0$, thus $\phi (\rho )$ is decreasing in $\rho$. Because $\rho \in [0,1]$, when $\rho =0$, $\phi (0)=\frac{2166{{a}^{2}}+3306ad-3483{{d}^{2}}}{69312}$ is a quadratic function of d. To ease calculation, we define $f(d)=2166{{a}^{2}}+3306ad-3483{{d}^{2}}$. By $f(0)>0$ and $f(1)>0$, we have $\phi (0)>0$ always holds when the expected random demand satisfies $\frac{d}{a}\in [0,\ \frac{1}{2}]$. When $\rho =1$, $\phi (1)=\frac{2166{{a}^{2}}+2166ad-9422{{d}^{2}}}{69312}$ is also a quadratic function of d. To ease calculation, we define $g(d)=2166{{a}^{2}}+2166ad-9422{{d}^{2}}$. Similarly, we can prove that $\phi (1)>0$ always holds when the expected random demand satisfies $\frac{d}{a}\in [0,\ \frac{1}{2}]$. Hence, $\phi (\rho )>0$ always holds when the expected random demand and the entry cost satisfy $\frac{d}{a}\in [0,\ \frac{1}{2}]$, $C\in ({{C}^{*}},\ \gamma (d))$ where ${{C}^{*}}$ is the unique solution to $\pi _{R}^{S}({{C}^{*}})=\pi _{R}^{N}({{C}^{*}})$, and $C\in (\gamma (0),\ \gamma (d))$. $\square$

Proof of Lemma 8

When the retailer does not share demand information, we have the manufacturer’s payoff difference between entry and no entry by calculating equation (32) as follows: $\pi _{M}^{NE}-\pi _{M}^{NN}= \frac{{{a}^{2}}(9{{\theta }^{3}}-52{{\theta }^{2}}+68\theta +32)+2ad[\theta \rho ({{\theta }^{2}}-14\theta +32)+2(4{{\theta }^{3}}-19{{\theta }^{2}}+18\theta +16)]+{{d}^{2}}[\theta {{\rho }^{2}}{{(4-\theta )}^{2}}+4\theta \rho (8-3\theta )+4(2{{\theta }^{3}}-8{{\theta }^{2}}+5\theta +8)]}{8({{\theta }^{3}}-8{{\theta }^{2}}+10\theta +16)} -C-\frac{\theta {{(2a+d+\rho d)}^{2}}}{8(1+\theta )}=\xi (d)-C$. Because $\frac{\partial \xi (d)}{\partial d}= \frac{(2-\theta )[2a({{\theta }^{3}}\rho -{{\theta }^{2}}\rho -6{{\theta }^{3}}+2{{\theta }^{2}}+26\theta +16)+2d(-{{\theta }^{2}}{{\rho }^{2}}+2{{\theta }^{3}}\rho -7{{\theta }^{3}}+2{{\theta }^{2}}+26\theta +16)]}{8(1+\theta )({{\theta }^{3}}-8{{\theta }^{2}}+10\theta +16)}>0$, we have the manufacturer will enter only when the entry cost C is lower than $\xi (d)$. $\square$

Proof of Lemma 9

When the retailer shares demand information, we have the manufacturer’s payoff difference between entry and no entry by calculating equation (34) as follows: $\pi _{M}^{SE}-\pi _{M}^{SN}= \frac{{{a}^{2}}(9{{\theta }^{3}}-52{{\theta }^{2}}+68\theta +32)+2a\delta [\theta \rho ({{\theta }^{2}}-14\theta +32)+2(4{{\theta }^{3}}-19{{\theta }^{2}}+18\theta +16)]+{{\delta }^{2}}[\theta {{\rho }^{2}}{{(4-\theta )}^{2}}+4\theta \rho (8-3\theta )+4(2{{\theta }^{3}}-8{{\theta }^{2}}+5\theta +8)]}{8({{\theta }^{3}}-8{{\theta }^{2}}+10\theta +16)} -C-\frac{\theta {{(2a+\delta +\rho \delta )}^{2}}}{8(1+\theta )}=\xi (\delta )-C$. By noting that $\delta \in [0,2d]$ and $\xi (\delta )$ is increasing in $\delta$, we have when $C\le \xi (0)$, the manufacturer will always enter; when $C>\xi (2d)$, the manufacturer will never enter; when C is between $\xi (0)$ and $\xi (2d)$, the manufacturer will enter only if $\delta >{\hat{\delta }}(C)$. $\square$

Proof of Lemma 10

In order to examine the retailers information sharing strategy, we derive the retailers payoff differences between information sharing and non-information sharing by calculating Eqs. (38) and (39) as follows: When $C\le \gamma (0)$, $\pi _{R}^{S}(C)-\pi _{R}^{N}(C)= \frac{{{(32a-12\theta a-2{{\theta }^{2}}a+{{\theta }^{3}}a+32\rho d-16\theta d+6{{\theta }^{2}}d-8{{\theta }^{2}}\rho d+{{\theta }^{3}}\rho d+4\theta \rho d)}^{2}}+16\theta {{(-4a+5\theta a-{{\theta }^{2}}a+4\theta d-{{\theta }^{2}}d-4\rho d+\theta \rho d)}^{2}}}{16{{({{\theta }^{3}}-8{{\theta }^{2}}+10\theta +16)}^{2}}}$ $+\frac{16\theta {{(4-\theta )}^{2}}{{(\theta -\rho )}^{2}}+{{(32\rho -16\theta +6{{\theta }^{2}}+4\theta \rho -8{{\theta }^{2}}\rho +\rho {{\theta }^{3}})}^{2}}}{48{{({{\theta }^{3}}-8{{\theta }^{2}}+10\theta +16)}^{2}}}{{d}^{2}}- \frac{{{(32a-12\theta a-2{{\theta }^{2}}a+{{\theta }^{3}}a+32\rho d-16\theta d+6{{\theta }^{2}}d-8{{\theta }^{2}}\rho d+{{\theta }^{3}}\rho d+4\theta \rho d)}^{2}}+16\theta {{(-4a+5\theta a-{{\theta }^{2}}a+4\theta d-{{\theta }^{2}}d-4\rho d+\theta \rho d)}^{2}}}{16{{({{\theta }^{3}}-8{{\theta }^{2}}+10\theta +16)}^{2}}}-\frac{(\theta +{{\rho }^{2}}){{d}^{2}}}{12}=$ $\frac{\theta [\rho ^2(-3\theta ^5+48\theta ^4-264\theta ^3+528\theta ^2-768) +\rho (12\theta ^4-160\theta ^3+560\theta ^2-256\theta -1024)-4\theta ^6+64\theta ^5 -320\theta ^4+420\theta ^3+688\theta ^2-1024\theta -1024]}{48*(\theta ^3-8*\theta ^2 +10*\theta +16)^2}$ $<0.$ When $C>\gamma (2d)$, $\pi _{R}^{S}(C)-\pi _{R}^{N}(C)=\frac{\theta {{(2\theta a+2\theta d+d-\rho d)}^{2}}+{{(2a+2\rho d+\theta \rho d-\theta d)}^{2}}}{16{{(1+\theta )}^{2}}}+\frac{\theta +4{{\theta }^{2}}-6\theta \rho +4{{\rho }^{2}}+\theta {{\rho }^{2}}}{48}{{d}^{2}} -\frac{\theta {{(2\theta a+2\theta d+d-\rho d)}^{2}}+{{(2a+2\rho d+\theta \rho d-\theta d)}^{2}}}{16{{(1+\theta )}^{2}}}-\frac{(\theta +{{\rho }^{2}}){{d}^{2}}}{12}=\frac{3\theta (\theta -1)+\theta (\rho ^2-6\rho +\theta )}{48}$. $\pi _{R}^{S}(C)-\pi _{R}^{N}(C)<0$ holds when $\theta <6\rho -\rho ^2$. Thus, the retailer will not share demand information with the manufacturer when the entry cost C is very low or especially high. $\square$