3. (24 %)

Consider a duopoly market that consists of two firms, OpenAI and Google. The competition proceeds as follows:

Stage 1: Each firm simultaneously determines its own R&D input level, $x_i$.

Stage 2: Knowing the R&D input level of each other, each firm simultaneously determines its own output level, $q_i$.

Each firm has the following cost function:

$$Ci(qi, xi, xj) = (70 - (xi + xj))qi + 2xi^2,$$

where $xj$ is the R&D level the other firm chooses. Let the inverse demand function of the market be $P = 100 - \frac{1}{3}Q$, where $Q = qO + q_G$.

(a) Consider the Cournot duopoly equilibrium. Given \( x{O}, x{G} \) determined, find the equilibrium output chosen by each firm in Stage 2.