Game Theoretic Analysis of Stackelberg Duopoly Competition

Suppose there are two firms, Lieder and Faulover, who are patent holders to produce a novel good. Since nobody else has a patent for the good, the firms would potentially form a duopoly (if they decide to produce). In order to start their productions, the firms must first build the factories, and the size of the factories will determine their outputs. Assume that once a factory is open, a firm commits to the quantity it would produce. That is, the firms are playing a Stackelberg duopoly game of sequential entry, and each firm’s choice of the factory size is that firm’s choice of output q_i (where i is either L for Lieder or F for Faulover). Lieder has obtained the patent early. Lieder’s (total) cost of production is C = 500 + 1.2(q_L)². Faulover has obtained its patent much later, and it is common knowledge that Lieder will finish its factory construction and start producing quantity q_L before Faulover makes the decision about its own production q_F. The delay with the patent, however, gives an opportunity for the Faulover’s engineers to develop several improvements to their technology that would lower the variable cost of production but increase the fixed cost (the added cost of R&D). If the engineers do make the improvements, Faulover’s (total) cost of production is expected to become C = 600 + 0.8(q_F)². If the engineers do not make the improvements, Faulover can pr