. (45 min) Consider two Örms, 1 and 2, who produce in a duopoly market. Firm 1 chooses its quantity of output x1 and Örm 2 chooses x2. Given these outputs, the market price is given by p = 1 (x1 + x2): There are zero costs of production. 1. Write the proÖt functions for Örms 1 and 2. Find the optimal quantity of output x2 for Örm 2, as a function of x1 (i.e. Önd Örm 2ís best response function). 2. DeÖne the notion of Nash equilibrium in pure strategies for this game (i.e. which conditions in terms of the choices xi and the proÖt function have to be satisÖed. 3. Now Önd the Nash equilibrium outputs for Örm 1 and Örm 2 and the attendant proÖts for each Örm. 4. Find the symmetric level of output (x1; x2) = (x; x) that would maximize the sum of Örm 1 and Örm 2ís proÖts if they each produce x . How does this compare to your answer from question [2.3], and why? In addition, explain clearly why producing x does not constitute a best response for either Örm. 5. Now return to the setting of question [2.3]. In the Nash equilibrium, Örm 2 chooses an output that is a best response to Örm 1ís output. Suppose that Örm 2 is concerned that it might be wrong about how much output 1 is going to produce, and so hires an industrial spy who can (reliably) report Örm 1ís output to Örm 2 before 2 makes a decision. Moreover, Örm 1 knows that Örm 2 has hired such a spy. As a result, Örm 1 knows that whatever value of x1 it produces, Örm 2 will produce a best response. Hence, from Örm 1ís point of view, x2is no longer to be viewed as Öxed when 1 maximizes 1ís proÖts, but rather x2 is e§ectively a function of x1, with this function given by Örm 2ís best response function from question [2.1]. 1. To analyze this situation, start with the proÖt function for Örm 1 from question [2.1], replace x2 in this proÖt function by Örm 2ís best response function, so that 1ís proÖts are now entirely a function of x1. Explicitly state the new proÖt function of Örm 1. 2. Now take the derivative of the proÖt function of Örm 1 with respect to x1 and solve to Önd the (new) equilibrium quantity x1. Insert this optimal quantity into Örm 2ís best response function to Önd Örm 2ís equilibrium quantity. 3. How do these compare to the quantities you found in question [2.3]? Explain the di§erences in your answers. Which Örm beneÖts from Örm 2ís spy and which one loses?
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