Entry of Innovator and License in Oligopoly

Abstract

When an outside innovating firm has a cost-reducing technology, it can sell licenses of its technology to incumbent firms using a combination of royalty and fixed fee. Alternatively, the innovating firm can enter the market and at the same time sell licenses, or enter the market without license. We examine the credibility of the threat of entry by the innovating firm using a two-step auction under oligopoly with three firms, one outside innovating firm and two incumbent firms. With general demand function, we show that the credibility of the two-step auction depends on the form of the cost function of the new technology, whether it is concave or convex. Also we analyze the optimal strategy for the innovator in a case of linear demand and quadratic cost functions in which the two-step auction is credible.

Appendices

Appendix A. Details of Calculations

$$ \begin{array}{@{}rcl@{}} \lambda_{A}&=& {c}_{A}^{2} {c}_{B}^{3} +4c_{A} {c}_{B}^{3} +4 {c}_{B}^{3} - {c}_{A}^{3} {c}_{B}^{2} +13c_{A} {c}_{B}^{2} +16 {c}_{B}^{2} -4 {c}_{A}^{3} c_{B}-10 {c}_{A}^{2} c_{B}+14c_{A}c_{B}\\ &&+25c_{B}-4 {c}_{A}^{3} -12 {c}_{A}^{2} +7c_{A}+14, \end{array} $$

$$ \begin{array}{@{}rcl@{}} \lambda_{B}&=&3{c}_{A}^{8} {c}_{B}^{4} + 42 {c}_{A}^{7} {c}_{B}^{4} + 236 {c}_{A}^{6} {c}_{B}^{4} + 684 {c}_{A}^{5} {c}_{B}^{4} + 1095 {c}_{A}^{4} {c}_{B}^{4} + 962 {c}_{A}^{3} {c}_{B}^{4} + 438 {c}_{A}^{2} {c}_{B}^{4} \\ && + 96c_{A} {c}_{B}^{4} + 8 {c}_{B}^{4} - 2 {c}_{A}^{9} {c}_{B}^{3} - 8 {c}_{A}^{8} {c}_{B}^{3} + 114 {c}_{A}^{7} {c}_{B}^{3} + 1012 {c}_{A}^{6} {c}_{B}^{3} + 3364 {c}_{A}^{5} {c}_{B}^{3} + 5696 {c}_{A}^{4} {c}_{B}^{3} \\ && + 5160 {c}_{A}^{3} {c}_{B}^{3} + 2424 {c}_{A}^{2} {c}_{B}^{3} + 552 c_{A} {c}_{B}^{3} + 48 {c}_{B}^{3} - 12 {c}_{A}^{9} {c}_{B}^{2} - 112 {c}_{A}^{8} {c}_{B}^{2} - 216 {c}_{A}^{7} {c}_{B}^{2} \\ && + 995 {c}_{A}^{6} {c}_{B}^{2} + 5454 {c}_{A}^{5} {c}_{B}^{2} + 10628 {c}_{A}^{4} {c}_{B}^{2} + 10296 {c}_{A}^{3} {c}_{B}^{2} + 5103 {c}_{A}^{2} {c}_{B}^{2} + 1230 c_{A} {c}_{B}^{2} + 114 {c}_{B}^{2} \\ && -24 {c}_{A}^{9} c_{B} - 256 {c}_{A}^{8} c_{B} - 896 {c}_{A}^{7} c_{B} - 700 {c}_{A}^{6} c_{B} + 2970 {c}_{A}^{5} c_{B} + 8444 {c}_{A}^{4} c_{B} + 9262 {c}_{A}^{3} c_{B}\\ && + 4964 {c}_{A}^{2} c_{B} + 1284 c_{A} c_{B} + 128 c_{B} - 16 {c}_{A}^{9} - 176 {c}_{A}^{8} - 688 {c}_{A}^{7} - 1060 {c}_{A}^{6} + 92 {c}_{A}^{5} + 2436 {c}_{A}^{4} \\ && + 3252 {c}_{A}^{3} + 1908 {c}_{A}^{2} + 528 c_{A} + 56, \end{array} $$

$$ \begin{array}{@{}rcl@{}} \lambda_{C} &=& {c}_{A}^{6} {c}_{B}^{6} + 12 {c}_{A}^{5} {c}_{B}^{6} + 54 {c}_{A}^{4} {c}_{B}^{6} + 112 {c}_{A}^{3} {c}_{B}^{6} + 105 {c}_{A}^{2} {c}_{B}^{6} + 36 c_{A} {c}_{B}^{6} + 4 {c}_{B}^{6} - {c}_{A}^{7} {c}_{B}^{5} + 74 {c}_{A}^{5} {c}_{B}^{5} \\ && + 406 {c}_{A}^{4} {c}_{B}^{5} +876 {c}_{A}^{3} {c}_{B}^{5} +826 {c}_{A}^{2} {c}_{B}^{5} +293c_{A} {c}_{B}^{5} +34 {c}_{B}^{5} -10 {c}_{A}^{7} {c}_{B}^{4} -53 {c}_{A}^{6} {c}_{B}^{4} +90 {c}_{A}^{5} {c}_{B}^{4} \\ &&+1125 {c}_{A}^{4} {c}_{B}^{4} +2716 {c}_{A}^{3} {c}_{B}^{4} +2641 {c}_{A}^{2} {c}_{B}^{4} +976c_{A} {c}_{B}^{4} +119 {c}_{B}^{4} -40 {c}_{A}^{7} {c}_{B}^{3} -264 {c}_{A}^{6} {c}_{B}^{3} \\ &&-338 {c}_{A}^{5} {c}_{B}^{3} +1258 {c}_{A}^{4} {c}_{B}^{3} +4179 {c}_{A}^{3} {c}_{B}^{3} +4372 {c}_{A}^{2} {c}_{B}^{3} +1705c_{A} {c}_{B}^{3} +220 {c}_{B}^{3} -80 {c}_{A}^{7} {c}_{B}^{2} \\ &&-552 {c}_{A}^{6} {c}_{B}^{2} -1148 {c}_{A}^{5} {c}_{B}^{2} +89{c}_{A}^{4} {c}_{B}^{2} +3210 {c}_{A}^{3} {c}_{B}^{2} +3928 {c}_{A}^{2} {c}_{B}^{2} +1650c_{A} {c}_{B}^{2} +227 {c}_{B}^{2} \\ && -80 {c}_{A}^{7} c_{B}-544 {c}_{A}^{6} c_{B}-1272 {c}_{A}^{5} c_{B}-852{c}_{A}^{4} c_{B}+1016 {c}_{A}^{3} c_{B}+1800 {c}_{A}^{2} c_{B}+840c_{A}c_{B}\\ && +124c_{B}-32 {c}_{A}^{7} -208 {c}_{A}^{6} -496 {c}_{A}^{5} -460{c}_{A}^{4} +32 {c}_{A}^{3} +324 {c}_{A}^{2} +176c_{A}+28, \end{array} $$

$$ \begin{array}{@{}rcl@{}} \lambda_{D} &=& 2(c_{B}+2)(c_{A}c_{B}+2c_{B}+2c_{A}+2)(c_{A}c_{B}+2c_{B}+2c_{A}+3)^{2}({c}_{A}^{2} c_{B}+4c_{A}c_{B}+c_{B}\\ &&+2 {c}_{A}^{2} +6c_{A}+2)^{2}, \end{array} $$

$$ \begin{array}{@{}rcl@{}} \lambda_{E} &=& {c}_{A}^{6} {c}_{B}^{4} +12 {c}_{A}^{5} {c}_{B}^{4} +54 {c}_{A}^{4} {c}_{B}^{4} +112 {c}_{A}^{3} {c}_{B}^{4} +105 {c}_{A}^{2} {c}_{B}^{4} +36c_{A} {c}_{B}^{4} +4 {c}_{B}^{4} - {c}_{A}^{7} {c}_{B}^{3} -4 {c}_{A}^{6} {c}_{B}^{3} \\ &&+34 {c}_{A}^{5} {c}_{B}^{3} +250 {c}_{A}^{4} {c}_{B}^{3} +584 {c}_{A}^{3} {c}_{B}^{3} +568 {c}_{A}^{2} {c}_{B}^{3} +205c_{A} {c}_{B}^{3} +24 {c}_{B}^{3} -6 {c}_{A}^{7} {c}_{B}^{2} -44 {c}_{A}^{6} {c}_{B}^{2} \\ &&-46 {c}_{A}^{5} {c}_{B}^{2} +347{c}_{A}^{4} {c}_{B}^{2} +1092 {c}_{A}^{3} {c}_{B}^{2} +1152 {c}_{A}^{2} {c}_{B}^{2} +448c_{A} {c}_{B}^{2} +57 {c}_{B}^{2} -12 {c}_{A}^{7} c_{B}-96 {c}_{A}^{6} c_{B}\\ &&-212 {c}_{A}^{5} c_{B}+88{c}_{A}^{4} c_{B}+871 {c}_{A}^{3} c_{B}+1062 {c}_{A}^{2} c_{B}+455c_{A}c_{B}+64c_{B}-8 {c}_{A}^{7} -64 {c}_{A}^{6} \\ &&-160 {c}_{A}^{5} -76{c}_{A}^{4} +254 {c}_{A}^{3} +384 {c}_{A}^{2} +182c_{A}+28, \end{array} $$

Appendix B. Stability Conditions

According to Seade (1980) and Dixit (1986), we consider stability conditions for a duopoly and an oligopoly. Consider the following matrix for a duopoly with Firm B and Firm C.

$$ \left[\begin{array}{cc} s_{B}\theta_{B}&s_{B}\sigma_{B}\\ s_{C}\sigma_{C}&s_{C}\theta_{C} \end{array}\right] $$

s_B and s_C are adjustment speeds of the outputs of Firms B and C. For the stability, the trace of this matrix must be negative, and its determinant must be positive. Therefore,

$$ s_{B}\theta_{B}+s_{C}\theta_{C}<0, $$

and

$$ s_{B}s_{C}[\theta_{B}\theta_{C}-\sigma_{B}\sigma_{C}]>0. $$

Since these are to hold independently of s_B and s_C, we need

$$ \theta_{B}<0,\ \theta_{C}<0,\ \theta_{B}\theta_{C}-\sigma_{B}\sigma_{C}>0. $$

Consider the following matrix for an oligopoly with Firms A, B, and C.

$$ \left[\begin{array}{ccc} s_{A}\theta_{A}&s_{A}\sigma_{A}&s_{A}\sigma_{A}\\ s_{B}\sigma_{B}&s_{B}\theta_{B}&s_{B}\sigma_{B}\\ s_{C}\sigma_{C}&s_{C}\sigma_{C}&s_{C}\theta_{C} \end{array}\right] $$

s_A, s_B, and s_C are adjustment speeds of the outputs of Firms A, B, and C. One necessary condition is that the trace of this matrix is negative. Therefore,

$$ s_{A}\theta_{A}+s_{B}\theta_{B}+s_{C}\theta_{C}<0. $$

Since this is to hold independently of s_A, s_B, and s_C, we need

$$ \theta_{A}<0, \theta_{B}<0,\ \theta_{C}<0. $$

Another necessary condition is that the determinant of this matrix has the sign of (− 1)³, that is, it is negative. Thus,

$$ \theta_{A}\theta_{B}\theta_{C}-\theta_{A}\sigma_{B}\sigma_{C}-\sigma_{A}\theta_{B}\sigma_{C}-\sigma_{A}\sigma_{B}\theta_{C}+2\sigma_{A}\sigma_{B}\sigma_{C}<0. $$

The second-order conditions in each case 𝜃_A < 0, 𝜃_B < 0, 𝜃_C < 0 with the stability conditions guarantee the existence of the locally unique stable Nash equilibrium. If the stability conditions are violated, the firms increase (or decrease) their outputs more than a change (increase or decrease) in the rival firm’s output, and then the outputs of the firms diverge from the equilibrium values. For the existence of the globally unique equilibrium, we need that 𝜃_A < 0, 𝜃_B < 0, 𝜃_C < 0 globally hold.