This is a Cournot competition:

Consider a market in which two firms are engage in quantity competition a la Cournot, but with differentiated products. As in the standard model each firm i=1,2 has a cost function TCi(qi)=F+cqi. However, now each firm may recieve a different price for it's output. In particular, firm 1 recieves the price P1=abq1dq2 and firm 2 recieves the price P2=abq2dq1. (a) Use the fact that MR1=P1+dq1dP1q1 to find an expression for MR1 in terms of a,b,d,q1 and q2. (b) Use your answer from part (a) to find firm 1's reaction function. (c) Find a simplified expression for each firm's equilibrium output, qi. (d) Find each firm's equilibrium price, Pi. Use your expression for Pi to find a simplified expression for Pic, the firms markup over marginal cost. (e) Find a simplified expression for firm i 's equilibrium profits, i. (f) Now consider your expressions for qi,Pi and i. How do these expression change if we start without differentiation (i.e. d=b ) and then move to a situation in which there is differentiation (i.e. d0, what would he choose? Explain. If firm 1 got to choose d2>0, what would he choose? Explain. Consider a market in which two firms are engage in quantity competition a la Cournot, but with differentiated products. As in the standard model each firm i=1,2 has a cost function TCi(qi)=F+cqi. However, now each firm may recieve a different price for it's output. In particular, firm 1 recieves the price P1=abq1dq2 and firm 2 recieves the price P2=abq2dq1. (a) Use the fact that MR1=P1+dq1dP1q1 to find an expression for MR1 in terms of a,b,d,q1 and q2. (b) Use your answer from part (a) to find firm 1's reaction function. (c) Find a simplified expression for each firm's equilibrium output, qi. (d) Find each firm's equilibrium price, Pi. Use your expression for Pi to find a simplified expression for Pic, the firms markup over marginal cost. (e) Find a simplified expression for firm i 's equilibrium profits, i. (f) Now consider your expressions for qi,Pi and i. How do these expression change if we start without differentiation (i.e. d=b ) and then move to a situation in which there is differentiation (i.e. d0, what would he choose? Explain. If firm 1 got to choose d2>0, what would he choose? Explain