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\( U\left(q_{1}, q_{2}\right)=\alpha\left(q_{1}+q_{2}\right)-\left(q_{1}^{2}+2 \gamma q_{1} q_{2}+q_{2}^{2}\right) / 2 \), 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\%)

(a) The inverse demand curve facing the two firms is given by the equation: P = ? - (q1 + q2) ? P=??(q1+q2) Where P is the price, q1 and q2 are the qu

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