Abstract
It is generally believed that the existence of gray channels hurts authorized retailers because gray marketers often free-ride on the marketing activities performed by authorized retailers. However, the effect on manufacturers’ profits is still rather vague. This paper sets up a two-stage sub-game perfect equilibrium model to examine the effects of gray goods on authorized retailers and manufacturers. It is found that manufacturers who are against parallel importation are likely to be those whose product has a low gray good penetration ratio, low price elasticity of demand, high cross-price elasticity of demand, or a high demand convexity.
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Notes
The mathematical details are available upon request from the author.
The marginal benefit of service is measured by p S x as shown in Eq. 6. Its value declines as the sales of the authorized product decrease.
The two axes in Fig. 1 represent the sale and service levels of the authorized retailer respectively. The slope of π x = 0 is defined by \({\raise0.7ex\hbox{${{\text{d}}S}$} \!\mathord{\left/ {\vphantom {{{\text{d}}S} {{\text{d}}x}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${{\text{d}}x}$}}\left| {_{\pi _x = 0} } \right. = - {\raise0.7ex\hbox{${\pi _{xx} }$} \!\mathord{\left/ {\vphantom {{\pi _{xx} } {\pi _{xS} }}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${\pi _{xS} }$}}\) which is positive as π xx < 0 and π xS > 0. Its intercept is \(x = {\raise0.7ex\hbox{${\omega - p}$} \!\mathord{\left/ {\vphantom {{\omega - p} {p_x }}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${p_x }$}} >0\) which can be derived by substituting S = 0 into the π x = 0 function to yields \(\pi _x \left| {_{S = 0} } \right. = p_x \left( {x;g,0} \right)x + p\left( {x;g,0} \right) - \omega = 0\). Similarly, the slope of π s = 0 is \({\raise0.7ex\hbox{${{\text{d}}S}$} \!\mathord{\left/ {\vphantom {{{\text{d}}S} {{\text{d}}x}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${{\text{d}}x}$}}\left| {_{\pi _s = 0} } \right. = - {\raise0.7ex\hbox{${\pi _{Sx} }$} \!\mathord{\left/ {\vphantom {{\pi _{Sx} } {\pi _{SS} }}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${\pi _{SS} }$}} >0\), and its intercept on the x axis is \(x = {\raise0.7ex\hbox{${f_S }$} \!\mathord{\left/ {\vphantom {{f_S } {p_S }}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${p_S }$}} >0\). Compare the two intercepts, we obtain: \({\raise0.7ex\hbox{${{\text{d}}S}$} \!\mathord{\left/ {\vphantom {{{\text{d}}S} {{\text{d}}x}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${{\text{d}}x}$}}\left| {_{\pi _x = 0} } \right. - {\raise0.7ex\hbox{${{\text{d}}S}$} \!\mathord{\left/ {\vphantom {{{\text{d}}S} {{\text{d}}x}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${{\text{d}}x}$}}\left| {_{\pi \; = 0} } \right. = {\raise0.7ex\hbox{${ - \pi _{xx} \pi _{SS} + \pi _{Sx}^2 }$} \!\mathord{\left/ {\vphantom {{ - \pi _{xx} \pi _{SS} + \pi _{Sx}^2 } {\pi _{xx} \pi _{xS} }}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${\pi _{xx} \pi _{xS} }$}} = {\raise0.7ex\hbox{${ - D}$} \!\mathord{\left/ {\vphantom {{ - D} {\pi _{xx} \pi _{xS} }}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${\pi _{xx} \pi _{xS} }$}} >0\). It indicates that the slope of π x = 0 is greater than that of π S = 0. Consequently, we have π x = 0 and π S = 0 curves as shown in Fig. 1.
As \({\raise0.7ex\hbox{${{\text{d}}x}$} \!\mathord{\left/ {\vphantom {{{\text{d}}x} {{\text{d}}x}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${{\text{d}}x}$}}\left| {_S = {\raise0.7ex\hbox{${ - p{}_x\left( {\bar S} \right)}$} \!\mathord{\left/ {\vphantom {{ - p{}_x\left( {\bar S} \right)} {\pi _{xx} }}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${\pi _{xx} }$}}} \right. <0\), an increase in the volume of the gray goods shifts the π x = 0 curve to the left.
The mathematical details are available upon request from the author.
The derivation of Eq. 12 is as follows: \(\frac{{{\text{d}}\Pi }}{{{\text{d}}g}} = \frac{{\partial \Pi }}{{\partial x}}\left[ {\frac{{\partial x}}{{\partial \omega }}\frac{{\partial \omega }}{{\partial g}} + \frac{{\partial x}}{{\partial g}}} \right] + \frac{{\partial \Pi }}{{\partial \omega }}\frac{{\partial \omega }}{{\partial g}} + \frac{{\partial \Pi }}{{\partial g}} = \left( {\omega - c} \right)\left( {x_\omega \omega _g + x_g } \right) + \left( {\omega - c} \right)\) (since \(\frac{{\partial \Pi }}{{\partial \omega }} = 0\))\( = \left( {\omega - c} \right)\left[ {x_\omega \frac{{ - \left( {x_g + 1} \right)}}{{\Pi _{\omega \omega } }} + \left( {x_g + 1} \right)} \right] = \left( {\omega - c} \right)\left( {x_g + 1} \right)\left( {\frac{{\Pi _{\omega \omega } - x_\omega }}{{\Pi _{\omega \omega } }}} \right) = \frac{1}{2}\left( {\omega - c} \right)\left( {x_g + 1} \right)\) (since \(\Pi _{\omega \omega } = 2x_\omega \)).
Given S, the profit function of the authorized retailer becomes of x only: π = π(x). Its first-order condition for profit maximization consists of Eq. 4 only. Totally differentiate this equation with respect to x, S and g yields the following comparative static effects: \(\frac{{{\text{d}}x}}{{{\text{d}}g}}\left| {_S } \right. = \frac{{ - \pi _{xg} }}{{\pi _{xx} }} = \frac{{ - \left( {1 + \alpha } \right)\left( {p_{xg} x + p_g } \right)}}{{p_{xx} x + 2p_x }} = \frac{{ - \left( {1 + \alpha } \right)^2 \left( {p_{xx} x + p_x } \right)}}{{p_{xx} x + 2p_x }}\) and \(\frac{{{\text{d}}x}}{{{\text{d}}S}} = - \frac{{\pi _{xS} }}{{\pi _{xx} }} = - \frac{{p_S }}{{\pi _{xx} }}\) .
The second-order condition for profit maximization requires.\(\pi _{xx} = p_{xx} x + 2p_x <0\) It is satisfied if \(\delta >- 2\).
This result can be explained by the following example. Imports of Benz Mercedes through a gray market would do more damages to the local price of the car than the same amount of imports through the authorized channel. Moreover, an increase in pirate goods would cause the price of the legitimate goods to plunge.
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Chen, HL. Gray Marketing: Does It Hurt the Manufacturers?. Atl Econ J 37, 23–35 (2009). https://doi.org/10.1007/s11293-008-9154-6
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DOI: https://doi.org/10.1007/s11293-008-9154-6