7.1. One-Shot Market Games The first group of problems asks you to find equilibria in one-shot Cournot and Bertrand Duopoly games. e Problem 7.1 Consider a duopoly market where the inverse market demand is specified by Q = 36 2P. The marginal cost of production is assumed to be zero for both firms. Calculate the Cournot equilibrium for a Cournot Duopoly game. o Problem 7.2 Derive a 2 x2 payoff matrix for the Cournot Duopoly game using the results of the first problem. Find the game's solution using the arrow diagram. o Problem 7.8 The Stackleberg equilibrium differs from Cournot's solution in that one firm gets to choose its output first. Hence, moves or output decisions are determined sequentially in a Stackleberg market. By way of contrast, Cournot's model assumed both firms made their output decisions simultaneously. How does a change from a simultaneous move problem to a sequential move market effect the firms' profit maximizing decisions? Suppose firm 1 chose output first. It is the Stackleberg leader. What output would it select? Firm 1 knows how firm 2, the Stackleberg follower, will behave because firm 1 knows firm 2's reaction function. Firm 1 will use this information to calculate its profit maximizing output. We use the market inverse demand function @ = 13 P and assume each firm has constant marginal costs equal to $1 per unit of output. Let , and , denote the first and second firms' outputs, respectively. Firm 1, the first mover, has a potential advantage because it can commit itself to a particular output decision before the second firm takes a decision. Is the rule giving firm 1 the first move financially advantageous? Do the profits of firm 1 rise compared to the corresponding Cournot equilibrium profits? The following steps should be followed to answer this question. (1) Begin by calculating each firms profit function in terms of both firms' outputs. (2) Firm 2 follows its Cournot reaction function. Why? Use it to arrive at a formula for firm 1's profit that is expressed entirely in its own output variable. (3) Compute firm 1's profit maximizing output using the profit function found in step 3. You can use calculus here or modify the geometric trick used to solve the Cournot model. Does the Dynamic Heuristic Principle help justify this step? (4) Calculate the second firm's output that is produced given your solution to item 4. Find the market price too. How do the Stackleberg equilibrium quantities and price compare to the corresponding Cournot equilibrium quantities and price? (5) Compute the firms' profits using this Stackleberg solution. Did profits rise or fall? (6) What do you conclude about the Stackleberg case how similar is this solution to the Cournot equilibrium? Can you think of a market where the Stackleberg story would provide interesting economic insights