Abstract
This paper develops an integrated duopoly model for coordinating R&D, product positioning, and pricing strategy. The model can be applied to a broad spectrum of market structures and market conditions since it allows for bidirectional technology transfers (i.e., firms can learn from each other), partial or complete technology transfer, differential quality-adjusted production costs across firms, preference heterogeneity across consumers for different product quality levels, and different behavioral modes of competitive reaction (e.g., sequential or simultaneous decision-making). We show that the managerial implications differ sharply for information goods and physical goods and vary depending on the behavioral modes chosen by the firms and on whether the technology transfers are unidirectional or bidirectional. Interestingly, contrary to common belief, for certain scenarios, product differentiation can increase when technology transfers are bidirectional.
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Notes
The results are independent of the sequence of quality introduction under zero marginal production cost (case 1), i.e., regardless of whether qualities are introduced simultaneously or sequentially in any order, strategic qualities are consistent with the results we show in case 1 where the low-end quality is a corner solution and the high-end quality is a constant.
The proof is trivial: \( \frac{d{p}_f}{d\theta }=\frac{2\overline{\tau}-\underline{\tau}}{3}\cdot \frac{d{q}_f}{d\theta }=0 \) and \( \frac{d{p}_h}{d\theta }=\frac{\overline{\tau}-2\underline{\tau}}{3}\cdot \frac{d{q}_h}{d\theta }=0 \).
Substituting the solutions of qualities back into the profit functions, we have: \( {\pi}^f=\frac{{\left(2\overline{\tau}-\underline{\tau}\right)}^2\left[{\left(2\overline{\tau}-\underline{\tau}\right)}^2-36{c}_f\underline{q}\right]}{324{c}_f}, \) \( {\pi}^h=\frac{{\left(\overline{\tau}-2\underline{\tau}\right)}^2}{9}\left[\frac{{\left(2\overline{\tau}-\underline{\tau}\right)}^2}{18{c}_f}-\underline{q}\right]-{\underline{q}}^2\left[{c}_h-\frac{\theta {\left(2\overline{\tau}-\underline{\tau}\right)}^2}{18{c}_f}\right] \). We see that firm f’s profit π f is unaffected by spillovers, \( \frac{d{\pi}^f}{d\theta }=0 \). But firm h’s profit is positively affected due to the reduction of its R&D costs via technology spillovers, \( \frac{d{\pi}^h}{d\theta }=\frac{\underline{q}{\left(2\overline{\tau}-\underline{\tau}\right)}^2}{18{c}_f}>0 \).
Oladi et al. [21] do not consider this cost scenario.
\( {\pi}_{11}^f=\frac{2{\beta}_f^2}{9{\left({q}_f-{q}_h\right)}^3}-2{c}_f \), \( {\pi}_{12}^f=-\frac{2{\beta}_f^2}{9{\left({q}_f-{q}_h\right)}^3}<0 \), \( {\pi}_{21}^h=-\frac{2{\beta}_f^2}{9{\left({q}_f-{q}_h\right)}^3}+2\theta {q}_h \), \( {\pi}_{22}^h=\frac{2{\beta}_f^2}{9{\left({q}_f-{q}_h\right)}^3}-2\left({c}_h-\theta {q}_f\right) \), π f13 = 0, and π h23 = 2q f q h > 0.
When there are no spillovers, the required condition is \( \frac{c_f{c}_h}{c_f+{c}_h}>\frac{\beta_f^2}{9\left({q}_f-{q}_h\right)} \). When there are spillovers, this condition is unnecessary.
To ensure that ∂q * f /∂q h > 1, the spillover parameter θ f has to be sufficiently large. Mathematically, the condition for θ f is \( \frac{36{c}_f{q}_h+{\left(2\overline{\tau}-\underline{\tau}\right)}^2-\left(2\overline{\tau}-\underline{\tau}\right){\left[{\left(2\overline{\tau}-\underline{\tau}\right)}^2+72{c}_f{q}_h\right]}^{\frac{1}{2}}}{36{q}_h^2}<{\theta}_f<\frac{c_f}{q_h}. \)
π f11 = − 2(c f − θ f q h ), π f12 = 2θ f q f > 0, π h21 = 2θ h q h > 0, \( {\pi}_{22}^h=-2{c}_h+\frac{{\left(2\overline{\tau}-\underline{\tau}\right)}^2{\left[{\left(\overline{\tau}-2\underline{\tau}\right)}^2{\theta}_f^2+9{\theta}_h{c}_f^2\right]}^{0.5}}{81{\left({c}_f-{\theta}_f{q}_h\right)}^3} \), π f13 = 2q f q h > 0, π f14 = 0, \( {\pi}_{23}^h=\frac{{\left(2\overline{\tau}-\underline{\tau}\right)}^2\left[{\left(\overline{\tau}-2\underline{\tau}\right)}^2\left({c}_f+{\theta}_f{q}_h\right)+9{\theta}_h{q}_h^2\left(3{c}_f-{\theta}_f{q}_h\right)\right]}{162{\left({c}_f-{\theta}_f{q}_h\right)}^3}>0 \), and \( {\pi}_{24}^h=\frac{q_f{\left(2\overline{\tau}-\underline{\tau}\right)}^2\left(2{c}_f-{\theta}_f{q}_h\right)}{18{\left({c}_f-{\theta}_f{q}_h\right)}^2}>0 \).
Mathematically, this requires \( {c}_h>\frac{\theta_f{\theta}_h{q}_f{q}_h}{c_f-{\theta}_f{q}_h}+\frac{{\left(2\overline{\tau}-\underline{\tau}\right)}^2\left[{\theta}_f^2{\left(\overline{\tau}-2\underline{\tau}\right)}^2+9{\theta}_h{c}_f^2\right]}{162{\left({c}_f-{\theta}_f{q}_h\right)}^3} \).
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Wang, X.(., Xie, Y., Jagpal, H.S. et al. Coordinating R&D, Product Positioning, and Pricing Strategy: A Duopoly Model. Cust. Need. and Solut. 3, 104–114 (2016). https://doi.org/10.1007/s40547-015-0063-y
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DOI: https://doi.org/10.1007/s40547-015-0063-y