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# (Solved): $$U\left(q_{1}, q_{2}\right)=\alpha\left(q_{1}+q_{2}\right)-\left(q_{1}^{2}+2 \gamma q_{1} q_{2}+q ... \( U\left(q_{1}, q_{2}\right)=\alpha\left(q_{1}+q_{2}\right)-\left(q_{1}^{2}+2 \gamma q_{1} q_{2}+q_{2}^{2}\right) / 2$$, where $$q_{i}, q_{i}$$ is the quantity of firm $$i$$ and firm $$j(i, j=1,2$$, and $$i \neq j), \alpha>0$$ and $$\gamma \in(0,1)$$ characterizes the degree of product differentiation. (a) Derive the inverse demand curve facing the two firms. (20\%) Suppose that the two firms start with the same constant marginal cost $$c(<\alpha)$$, that the firms have identical $$R \& D$$ technologies and either firm can lower its marginal cost by $$x_{i}$$ after spending an amount $$V\left(x_{i}\right)=v x_{i}^{2} / 2$$ on $$R \& D$$, and that, with spillovers, the firms' post innovation marginal costs are given by $$c_{i}=c-x_{i}-\beta x_{j}$$, $$i, j=1,2$$, and $$i \neq j, \beta \in(0,1)$$ captures the extent of spillovers. In a two stage framework with firms first undertake R\&D investment, then compete a la Cournot in the product market. (b) Investigate the conditions under which the Cournot-Nash equilibrium sustains in the output game. $$(15 \%)$$ (c) Examine the conditions under which the Cournot-Nash equilibrium in the R\&D game sustains. (15\%)

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