= Consider a market in which two firms are engage in quantity competition a la Cournot,...

 

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= 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 + c* q. However, now each firm may recieve a different price for it's output. In particular, firm 1 recieves the price P₁ = a - b * q1 d* q2 and firm 2 recieves the price P₂=a-b* q2 - d * 91. dP₁ (a) Use the fact that MR₁ = P₁ + * 9₁ to find an expression for MR₁ in terms of a, b, d, q1 dq₁ and 92. (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, q. (d) Find each firm's equilibrium price, P*. Use your expression for P to find a simplified expression for P* – c, the firms markup over marginal cost. (e) Find a simplified expression for firm i's equilibrium profits, π. (f) Now consider your expressions for q, P and T. 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. d<b?) What intuition can you give for this result? (g) Now consider a more complicated version of the model in which P₁ = a-b* q1 − d₁ * q2 and P₂ = ab* q2 - d2 * 91. Solve for each firm's reaction function in this case. (h) Do NOT solve for the equilibrium of the model from (g), but use the reaction functions you found in (g) to answer the following question. If firm 1 got to choose d₁ > 0, 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 + c* q. However, now each firm may recieve a different price for it's output. In particular, firm 1 recieves the price P₁ = a - b * q1 d* q2 and firm 2 recieves the price P₂=a-b* q2 - d * 91. dP₁ (a) Use the fact that MR₁ = P₁ + * 9₁ to find an expression for MR₁ in terms of a, b, d, q1 dq₁ and 92. (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, q. (d) Find each firm's equilibrium price, P*. Use your expression for P to find a simplified expression for P* – c, the firms markup over marginal cost. (e) Find a simplified expression for firm i's equilibrium profits, π. (f) Now consider your expressions for q, P and T. 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. d<b?) What intuition can you give for this result? (g) Now consider a more complicated version of the model in which P₁ = a-b* q1 − d₁ * q2 and P₂ = ab* q2 - d2 * 91. Solve for each firm's reaction function in this case. (h) Do NOT solve for the equilibrium of the model from (g), but use the reaction functions you found in (g) to answer the following question. If firm 1 got to choose d₁ > 0, what would he choose? Explain. If firm 1 got to choose d2 > 0, what would he choose? Explain.

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A a The marginal revenue MR1 for firm 1 is given by MR1 P1 P1q1 a bq1 dq2 b a bq1 dq2 b a b bq1 dq2 b Using the expression for MR1 above firm 1s react... View the full answer