econ quiz

1. Consider an exchange economy with two goods, .1: and y, and two agents, 1 and 2. Each agent is initially endowed with 5 units of 3: and 5 units of 3;. Agent 2 has preferences for .1: and y represented by the utility function H3(Ig,'y2} = Igyg. Agent 1 also likes a\" and 3;, but, in particular, it derives utility from consuming more .5: than agent 2. Agent 1's preferences are represented by the utilit;r function arl, pl, 332] = (5:1 1:2}19'1. Thus, while it does not choose 3:2, it is affected by it. a. Assuming the agents were able to trade at competitively determined prices, set up and solve each agent 's maximization problem and determine its demands for :r and y. b. Find a competitive equilibrium. c. Explain how the process of determining a competitive equilibrium in this case differs from the usual case in which agents only care about their own consumption. d. Set up and solve the Social Planner's Problem for determining the Pareto eicient allocations. e. Verify that the equilibrium allocation in part b is inefcient. Explain why. f. |Compare the above outcome in b to the case in which agent 1 had \"standard\" preferences and only cared about its own consumption, say, 11.1 [31, pl} 2 mm. What would be the equilibrium allocation in that case?' Does the fact that 1 compares its a: consumption to 2's increase or decrease the amount 1 demands of good I? g. Argue that if there were \"mutual one-upmanship,\" that is, if each tried to consume more than the other (say, tidal-ns?) 2 {I33 rjlyi, for 2,}: = 1,2, 2' 75 3'}, then no competitive equilibrium would exist in such an economy. 2. A rmJ E3, is considering entering a market with a single incumbent rm? I. If E entersj rm I can either ght or share the market. Before E enters} I thinks that with probability p3, E 1s technology is better than I 1s and with probability 1 p, E 's technology is worse {where U :2 j} a: 1]. When it considers entering, E knows if its technology is better or worse than I 1s [the result of past choices that it cannot change}. If E does not enter} then it gets [I prot and I gets a prot of ID. If E enters with worse technology and I ghts, then E gets prot of 4 and I gets 5. If1 instead? I shares the market? then E gets prot of 1 and I gets 4. If E enters with better technology and I ghts1 then E gets prot of 1 and I gets 3. If E enters with better technology and I shares the market1 then E gets prot of 4 and I gets 1. Each rm seeks to maximize its expected prot. The information above is common knowledge. The rms' interaction can be represented by the game tree in Figure 1. FIGURE 1 a. At each node in the tree in Figure 13. write a label for the player who moves at that node. Label each move in the tree and write the rms' payoffs at the appropriate places. b. How man},r pure strategies does rm E have? Give an example of one of them. c. Find the set of Nash equilibria {NE} in which I chooses a pure strategy. Describe the NE strategies. Does this set of NE depend on the value of 3:? If so, how?I If not, show that it does not. In the remaining parts of the problemJ assume that p = 1,32. d. Find every sequential equilibrium {SE} in which rm I chooses a pure strategy. e. If an equilibrium existed in which I chose a totally mixed strategy? how could this behavior by I be interpreted as part of its interaction with rm E? f. Does the game have a sequential equilibrium {SE} in which I chooses a totallyr mixed strategy? If so1 nd one. If not. show that no such SE exists. g. lCompare the SE's you have found. Are any of them more plausible than the oth ers as representations of outcomes of the interaction between rational rms with the characteristics of E and I? Describe and explain the relevant differences among these SE's