Sequencing R&D decisions with a consumer-friendly firm and spillovers

Abstract

This study considers a duopoly model in which both a consumer-friendly (CF) firm and a for-profit (FP) firm undertake cost-reducing R&D investments in an endogenous R&D timing game and then play Cournot output competition. When the CF firm chooses its profit-oriented consumer-friendliness, we show that the consumer-friendliness is non-monotone in spillovers under both simultaneous move and sequential move with FP firm’s leadership while it is decreasing under sequential move with CF firm’s leadership. We also show that a simultaneous-move outcome is a unique equilibrium when the spillovers are low and the CF firm invests higher R&D and obtains higher profits. When the spillovers are not low, two sequential-move outcomes appear and the CF firm might obtain lower profits with higher spillovers under the CF firm leadership.

Appendices

Appendix A. Simultaneous and sequential R&D competition

Proof of Lemma 1

$$\begin{aligned} x_{0}- x_{1}&=\frac{3 (a-c) (5-\theta ) \theta (29+19 \beta -7 \theta -5 \beta \theta )}{\Delta _1}\ge 0, \quad \text {for any} \; \theta \in [0,1] \; \text {and} \; \beta \in [0,1]\\ \frac{\partial x_0}{\partial \theta }&=\frac{3 (a-c) \epsilon _1}{\Delta _1^2}>0, \qquad \frac{\partial x_1}{\partial \theta }=-\frac{12 (a-c) \epsilon _2}{\Delta _1^2}<0. \end{aligned}$$

where

$$\begin{aligned}&\epsilon _1\equiv 163995-141056 \theta +45131 \theta ^2-6394 \theta ^3+340 \theta ^4\\&\quad -4 \beta ^5 \left( -100-48 \theta +91 \theta ^2-30 \theta ^3+3 \theta ^4\right) \\&\quad -4 \beta ^4 \left( 1480-720 \theta +127 \theta ^2-22 \theta ^3+3 \theta ^4\right) \\&\quad +\beta ^3 \left( -3240-3088 \theta +3023 \theta ^2-694 \theta ^3+47 \theta ^4\right) \\&\quad +\beta ^2 \left( 105500-95904 \theta +32855 \theta ^2-5030 \theta ^3+291 \theta ^4\right) \\&\quad +\beta \big (257665-224184 \theta +72719 \theta ^2-10482 \theta ^3 +570 \theta ^4\big )>0 \end{aligned}$$

and

$$\begin{aligned}&\epsilon _2\equiv 21025-19488 \theta +6749 \theta ^2-1038 \theta ^3+60 \theta ^4\\&\quad -2 \beta ^5 (-4+\theta )^2 \left( 15-2 \theta +\theta ^2\right) \\&\quad +\beta ^2 \left( -19665+17192 \theta -5482 \theta ^2+752 \theta ^3-37 \theta ^4\right) \\&\quad +\beta ^3 \left( -3100+4596 \theta -2170 \theta ^2+424 \theta ^3-30 \theta ^4\right) \\&\quad +\beta ^4 \left( 3940-3024 \theta +725 \theta ^2-42 \theta ^3-3 \theta ^4\right) \\&\quad +\beta \left( 6380-8220 \theta +3656 \theta ^2-692 \theta ^3+48 \theta ^4\right) >0. \end{aligned}$$

$\square$

Proof of Proposition 1

$\frac{\partial ^2 \pi _0}{\partial \theta ^2}<0$ for any $\theta \in [0,1]$ and $\beta \in [0,1]$. Now $\frac{\partial \pi _0}{\partial \theta }\big |_{\theta \rightarrow 0}=\frac{30 (a-c)^2 \left( 2871-4256 \beta +816 \beta ^2+2784 \beta ^3-784 \beta ^4\right) }{\left( 29+20 \beta -4 \beta ^2\right) \left( 59-12 \beta +4 \beta ^2\right) ^3}>0$ for any $\beta \in [0,1]$ and $\frac{\partial \pi _0}{\partial \theta }\big |_{\theta \rightarrow 1}=-\frac{(a-c)^2 \left( 157197+399959 \beta +259650 \beta ^2-33382 \beta ^3-37981 \beta ^4+17615 \beta ^5+1512 \beta ^6-2240 \beta ^7+294 \beta ^8\right) }{3 \left( 71+26 \beta -25 \beta ^2+4 \beta ^3\right) ^3}<0$ for any $\beta \in [0,1]$. The fact that $\frac{\partial \pi _0}{\partial \theta }\big |_{\theta \rightarrow 0} >0$ and $\frac{\partial \pi _0}{\partial \theta }\big |_{\theta \rightarrow 1}<0$ implies the existence of $\theta ^{sm}\in (0,1)$ such that $\frac{\partial \pi _0}{\partial \theta }=0$.

By substituting $\theta ^{sm}$ into $q_i$, $x_i$ and $\pi _i$ we obtain the Fig. 1. $\square$

Fig. 1

Equilibrium in a simultaneous R&D

Proof of Lemma 2

$$\begin{aligned} x_0- x_1= -\frac{(a-c) \epsilon _3}{\Delta _2} \end{aligned}$$

where $\epsilon _3\equiv 192-4925 \theta +3402 \theta ^2-798 \theta ^3+63 \theta ^4+\beta ^3 \left( -768+1292 \theta -543 \theta ^2+75 \theta ^3-2 \theta ^4\right) +3 \beta \big (-528-801 \theta +906 \theta ^2-268 \theta ^3+25 \theta ^4\big )+\beta ^2 \big (3456-3036 \theta +1443 \theta ^2-369 \theta ^3+36 \theta ^4\big )<0$ iff ${\hat{\theta }}<\theta \le 1$ for any $\beta \in [0,1]$. $\square$

Regarding $\frac{\partial x_0}{\partial \theta }$ and $\frac{\partial x_1}{\partial \theta }$ we show the following contour-plots: See Fig. 2

Fig. 2

Derivative of x_i w.r.t θ in a sequential scenario with FP leadership

Proof of Proposition 2

Let $\mu _1\equiv \scriptstyle 44627383183+27162552896 \beta -97878998532 \beta ^2+16174246080 \beta ^3+137506521008 \beta ^4+78638538496 \beta ^5+70604951232 \beta ^6-4968843264 \beta ^7-4994814720 \beta ^8+8267472896 \beta ^9-5900045312 \beta ^{10}+1783873536 \beta ^{11}-395538432 \beta ^{12}+70451200 \beta ^{13}-5619712 \beta ^{14}>0$ and $\mu _2\equiv \scriptstyle 44267381717+131915016701 \beta +70664147757 \beta ^2-64122805393 \beta ^3-15190851132 \beta ^4+16292528946 \beta ^5-14273759238 \beta ^6+8826622830 \beta ^7 +115135101 \beta ^8-2259056223 \beta ^9+625964841 \beta ^{10}-15435117 \beta ^{11}-12286566 \beta ^{12}-419904 \beta ^{13}>0$. Then,

$\frac{\partial ^2 \pi _0}{\partial \theta ^2}<0$ for any $\theta \in [0,1]$ and $\beta \in [0,1]$. Now $\frac{\partial \pi _0}{\partial \theta }\big |_{\theta \rightarrow 0}=\frac{2 (a-c)^2 \mu _1}{\left( 73+54 \beta -36 \beta ^2+8 \beta ^3\right) ^3 \left( 249+10 \beta +28 \beta ^2-8 \beta ^3\right) ^3}>0$ for any $\beta \in [0,1]$ and $\frac{\partial \pi _0}{\partial \theta }\big |_{\theta \rightarrow 1}=-\frac{(a-c)^2 \mu _2}{\left( 143-18 \beta +15 \beta ^2\right) ^3 \left( 47+30 \beta -33 \beta ^2\right) ^3}<0$ for any $\beta \in [0,1]$. The fact that $\frac{\partial \pi _0}{\partial \theta }\big |_{\theta \rightarrow 0} >0$ and $\frac{\partial \pi _0}{\partial \theta }\big |_{\theta \rightarrow 1}<0$ implies the existence of $\theta ^{fp}\in (0,1)$ such that $\frac{\partial \pi _0}{\partial \theta }=0$.

By substituting $\theta ^{fp}$ into $q_i$, $x_i$ and $\pi _i$ we obtain the Fig. 3. $\square$

Fig. 3

Equilibrium in a sequential R&D with FP leadership

Proof of Lemma 3

$$\begin{aligned} x_{0}- x_{1}=\frac{(a-c) \epsilon _4}{\Delta _3}\ge 0, \quad \text {for any} \; \theta \in [0,1] \; \text {and} \; \beta \in [0,1] \end{aligned}$$

where $\epsilon _4\equiv 192+4253 \theta -2637 \theta ^2+531 \theta ^3-35 \theta ^4+\beta ^3 \left( -768+100 \theta +72 \theta ^2-12 \theta ^3\right) +\beta \big (-1584+6111 \theta -3435 \theta ^2 +681 \theta ^3-45 \theta ^4\big )-12 \beta ^2 \left( -288+116 \theta +13 \theta ^2-10 \theta ^3+\theta ^4\right) \ge 0$. $\square$

Regarding $\frac{\partial x_0}{\partial \theta }$ and $\frac{\partial x_1}{\partial \theta }$ we show the following contour-plots: See Fig. 4

Fig. 4

Derivative of x_i w.r.t θ in a sequential scenario with CF leadership

Proof of Proposition 3

Let $\scriptstyle \mu _3\equiv 2230767+898952 \beta -4288472 \beta ^2-1749792 \beta ^3+669920 \beta ^4-324608 \beta ^5+8832 \beta ^6+32768 \beta ^7-4352 \beta ^8>0$ iff $\beta <0.7364$; and $\scriptstyle \mu _4\equiv 61252038+207486522 \beta +222060609 \beta ^2+74691882 \beta ^3+3472929 \beta ^4-9569790 \beta ^5-8218953 \beta ^6+4566642 \beta ^7-2702812 \beta ^8+967840 \beta ^9-74729 \beta ^{10} -9340 \beta ^{11}-6715 \beta ^{12}+2564 \beta ^{13}-223 \beta ^{14}>0$. Then,

$\frac{\partial ^2 \pi _0}{\partial \theta ^2}<0$ for any $\theta \in [0,1]$ and $\beta \in [0,1]$. Now $\frac{\partial \pi _0}{\partial \theta }\big |_{\theta \rightarrow 0}=\frac{2 (a-c)^2 \mu _3}{\left( 18177+14176 \beta -6380 \beta ^2+2560 \beta ^3-1360 \beta ^4+512 \beta ^5-64 \beta ^6\right) ^2}>0$ iff $\beta <0.7364$ and $\frac{\partial \pi _0}{\partial \theta }\big |_{\theta \rightarrow 1}=-\frac{(a-c)^2 \mu _4}{4 \left( 477+324 \beta -264 \beta ^2+24 \beta ^3-20 \beta ^4+18 \beta ^5-3 \beta ^6\right) ^3}<0$ for any $\beta \in [0,1]$. The fact that $\frac{\partial \pi _0}{\partial \theta }\big |_{\theta \rightarrow 0} >0$ and $\frac{\partial \pi _0}{\partial \theta }\big |_{\theta \rightarrow 1}<0$ implies the existence of $\theta ^{cf}\in (0,1)$ such that $\frac{\partial \pi _0}{\partial \theta }=0$.

By substituting $\theta ^{cf}$ into $q_i$, $x_i$ and $\pi _i$ we obtain the Fig. 5. $\square$

Fig. 5

Equilibrium in a sequential R&D with CF leadership

Appendix B. Values of $\Delta _i$ and $\nu _i$

$$\begin{aligned}&\Delta _1\equiv 1711+832 \beta -360 \beta ^2+128 \beta ^3-16 \beta ^4\\&\quad +\left( -1064-680 \beta +27 \beta ^2-112 \beta ^3+20 \beta ^4\right) \theta \\&\quad +\left( 220+174 \beta +39 \beta ^2+36 \beta ^3-4 \beta ^4\right) \theta ^2\\&\quad +\left( -15-14 \beta -6 \beta ^2-4 \beta ^3\right) \theta ^3>0.\\&\Delta _2\equiv \left( 73+54 \beta -36 \beta ^2+8 \beta ^3+\left( -29-28 \beta +\beta ^2-10 \beta ^3\right) \theta \right. \\&\quad \left. +\left( 3+4 \beta +2 \beta ^2+2 \beta ^3\right) \theta ^2\right) \cdot \big (249+10\beta \\&\quad +28 \beta ^2-8 \beta ^3+\left( -121-36 \beta -15 \beta ^2+10 \beta ^3\right) \theta \\&\quad +\left( 15+8 \beta +2 \beta ^2-2 \beta ^3\right) \theta ^2\big )>0.\\&\Delta _3\equiv 18177+14176 \beta -6380 \beta ^2+2560 \beta ^3-1360 \beta ^4+512 \beta ^5\\&\quad -64 \beta ^6+\big (-14149-12496 \beta +1897 \beta ^2\\&\quad -3552 \beta ^3+1432 \beta ^4-256 \beta ^5+16 \beta ^6\big ) \theta \\&\quad +\left( 4104+4046 \beta +444 \beta ^2+1688 \beta ^3-432 \beta ^4+32 \beta ^5\right) \theta ^2\\&\quad +\big (-525-572 \beta -203 \beta ^2-336 \beta ^3+40 \beta ^4\big ) \theta ^3\\&\quad +\left( 25+30 \beta +18 \beta ^2+24 \beta ^3\right) \theta ^4>0.\\&\nu _1\equiv \left( -1263-373 \beta +2776 \beta ^2-1476 \beta ^3+928 \beta ^4-160 \beta ^5\right) \theta \\&\quad +\left( -686-1330 \beta -1507 \beta ^2+175 \beta ^3-420 \beta ^4+132\beta ^5\right) \theta ^2\\&\quad +\big (282+500 \beta +429 \beta ^2+77 \beta ^3 +72 \beta ^4-40 \beta ^5\big ) \theta ^3\\&\quad +\left( -27-51 \beta -44 \beta ^2-14 \beta ^3-4 \beta ^4+4 \beta ^5\right) \theta ^4.\\&\nu _2\equiv \left( -1119+2443 \beta +1852 \beta ^2-1272 \beta ^3+272 \beta ^4-16 \beta ^5\right) \theta +\left( -561-2099 \beta -1360 \beta ^2+416 \beta ^3-32 \beta ^4\right) \theta ^2\\&\quad +\left( 191+469 \beta +316 \beta ^2-40 \beta ^3\right) \theta ^3-3 \big (5+11 \beta +8 \beta ^2\big ) \theta ^4. \end{aligned}$$