Graduate school Industrial Organization Assignment.

Collusion& Dynamic games

Assignment 4 1. Consider an infinitely repeated game between 2 firms that produce a homogeneous commodity. Firms face the following aggregate (inverse) demand curve for the product: p = a - bq. Firms face zero costs. The discount rate is denoted by 8 [0,1]. a. Assume that firms follow a grim trigger strategy whereby if at any period either firm deviates from the collusive quantity, both firms will revert to the Cournot outcome forever. Also, assume that in the collusive equilibrium both firms share the market equally. Show that collusion is a feasible equilibrium of this game, as long as firms are sufficiently patient. Show what that minimum patience level is (i.e. the critical discount factor). b. Use your answer to a. to intuitively illustrate the multiplicity of equilibria predicted by the Folk theorem. c. Consider now a variant of this exercise in the spirit of the model developed by Rotenberg and Saloner (1986). Specifically, with probability 0.1, the demand curve in any given period will experience a positive shock and would look like: p = 2a - b; these are considered "high demand" periods. The rest of the time (i.e. with probability 0.9), the demand curve will be the one given at the beginning of this problem; these are considered "low demand periods. As in the original RS model, firms learn with 100% certainty the demand realization for the current period, but remain uncertain about future demand realizations (i.e. they only know their distribution). Demonstrate that for a firm with discount factor equal to 0.6, collusion is only sustainable during low demand period but not during a high demand period. (Hint: the RHS of the incentive compatibility constraint now has an expectation while the LHS does not). d. Is your answer to c. consistent with the "countercyclical" collusion prediction in the RS model? Why or why not? 2. This problem illustrates the positive) effect of multimarket contact on the feasibility of collusion Consider an infinitely repeated Bertrand game with homogeneous products between two companies. As opposed to the base case we covered in class, these two companies (1.2) operate in two markets (A,B) and are asymmetric in size. Specifically, firm 1 is larger than fimm 2 in market A and vice versa. Assume that this asymmetry is "symmetric": in market A firm l's market share is y and firm 2's market share is 1 - y, and the opposite occurs in market B (where y > 1/2). Assuming that firms play a grim trigger strategy and that when colluding they share the monopoly profits according to their market shares, show that: a. When considering collusion as a possible Nash equilibrium of the infinitely repeated game in isolation (i.e. analyzing each market separately), the critical discount factor in either market) is lower for the firm with a larger market share than for the firm with the smaller market share. According to this result, for which of the two firms is collusion more stable (i.e., the ICC is more easily satisfied)? Of the two critical discount factors you obtained (one for each firm), which is binding in order for collusion to be feasible in each market (when studied in isolation)? b. When pooling both IC constraints (firms care about their overall well-being and the overall possibility to sustain collusion), show that collusion possibilities increase since the firm for which collusion was less stable in a. (markets in isolation) can now more easily sustain collusion (i.e. the binding critical discount factor in a goes down). Also, c. Why would the possibilities of collusion (as measured by the critical discount factor) be exactly the same in the no-multimarket contact case and the multimarket contact case if firms were identical (i.c. y = 1/2) in both markets? d. Using your answers in a.-c., provide the economic intuition for why multimarket contact, in the way described, can make (tacit) collusion more likely than in a market where multimarket contact is non-existent